Stay in bodyguard mode! (defensive follow/stay)
Bodyguard provides amazing protection with the damage split between all your pets. It can be dangerous and less effective in the early levels but it still works pretty well.
A solution that is non-optimal, but simple, is to edit two out of your three initial pet control buttons, the ones that show up in your power tray. Edit the "follow" button so that you switch the mode from "Aggressive" to "Defensive", which puts your pets into bodyguard mode every time you press that button. Second, edit the "Attack" button as well, changing "Aggressive" to "Defensive" once again. That tells your pets to attack the mob you have targeted until it is defeated. You lose bodyguard mode while your pets are carrying out that attack, but they automatically switch back to bodyguard mode after they defeat that mob.
Dark Miasma is a very useful Secondary in these types of situations. With a soft hold in Fearsome Stare and a nice Slow in Tar Patch, usually you can get off a few heals with Twilight Grasp for you and your pets. And if that's not enough, the secondary effects of those powers lower the enemy's damage resistance, to-hit chances and regeneration rate... making them a lot easier to kill. 
Move out of the burn patch and stay in Bodyguard mode. Also don't run to the bank. Mayhem missions are easy if you take it slow and walk to the bank, killing stuff for more time. In fact you should read the wiki page on Mayhem Missions. and Bodyguard mode
Captain Den'Rath 53* Merk/Traps MM, Rivona 50Energy Blast/Time Cor,Victoria Von Heilwig 53* Dual Pistols/Traps Cor, Crab Spider Webguard 53* SOA, Accela 53* Bot/FF MM,Valkyrie's Executor 53* Broadsword/Shield Def Scrap. On FREEDOM! @Knight Of Bronze
"Hypocrisy, the human inherent." "Let not this work be wasted, apply yourself always."
Your strategy may well depend on your primary/secondary combination. Or, maybe that's not really necessary for me to state 
The "unshakable agro" has always annoyed me, but then, it's just something to work around - it is what it is.
I will default to BG if I have to, but I like to at least TRY to get some focused damage going first, before I have to "turtle up."
I've had great luck primarily with /traps and /dark. Both work wonderfully, /traps the best, for mucking up the boss's pathing or simply giving yourself a little time. Basically, just find a corner you can target around but can use to deny LOS to the boss (and LB entourage). Drop caltrops or tar patch, as applicable, to slow the responders. However, before you drop the GTAOE, put your pets on "goto agro" and place them where you know the ambush is coming from, so they have to walk THROUGH your pets INTO your slow patch. That'll often give me enough time to create some sort of effect to shape the battle.
Different banks (and ambush locations, but I worry *the most* about the actual boss, especially if it's a high damage level 50+ boss) have different maps with different opportunities. Ideally, I like to be around a "U" shaped corner, which often will make the boss stop, think, and get "confused" on the GTAOE patch. Every second he hesitates is another chance to kill a LB or put more damage/effect on the boss.
Necro/ is good thanks to the minor -tohit most of their attacks apply, and lich's holds. Bots/ of course is great because of the KB, but of course some bosses are quite resistant to the KB effect. My Ninja/ is ... OK. At least they put out damage the fastest, but since I run Ninja/FF there's no slow function. I don't find a great deal of synergy with repulsion and force bubble and melee pets, so I don't usually run them. Rather, if I need to tactically retreat to another corner, I'll force bolt the boss or LB off me, and focus pet fire on that thing while I back off. If I have to I'll turtle up in BG, and if I REALLY have to I'll PFF up... but I'll usually put the pets in agro mode prior, not BG since it can be a bit wonky IME when I'm PFF'd. If necessary I'll use detention field to snag the random LB that gets through, but I am more annoyed than enthused by that power.
My necro/ is /dark, so in these engagements I have pretty good luck - tar patch usually slows the creeps down enough that fluffy and lich can hold/slow most of them such that not many make it through. And I can at least try to save myself with twilight grasp if someone makes it far enough to smack me.
Often if I can slow the boss enough I'll focus fire deliberately on the LB, especially the sneaky ones that can do the most quick damage, and then I'll push on to the boss.
I honestly have found it hit or miss to use a big debuff like darkest night. First, in order to apply it I'd really have to be in BG mode, and I'm still vulnerable to the boss's first attack. I prefer to try to stay out of BG unless I need it, so I really don't start bank boss fights with it. I'll use it instead in regular doors. But, starting in BG and applying darkest night (or the various equivalent debuffs in /poison, /thermal etc) is a perfectly good strategy - I just find it to be slower and bit more random. YMMV of course.
I don't have much experience with /thermal... I have run it into the low 20s, no more. It would seem you would simply have to spend a lot more time in /bg, using your heals to keep pets alive as damage was shared, and hoping warmth would keep you alive. Heat exhaustion would seem to be the power of choice for slowing down incoming damage? Again, my gut feeling here is that BG is probably going to be the mode of choice for most of the fights in a bank.
/Storm is kind of painfully easy, truth be told. I have an "easy-mode" bots/storm... I hate the look of hurricane, but will use it specifically (and generally only) for banks. I'll just throw on 'cane, stand right around a corner with the pets out in agro mode, so the boss'll have to run through the pets into 'cane. When he's just approaching pets I'll drop freezing rain. Every once in a while I'll get unlucky and get one-shotted but it's pretty rare.
I did run a couple of mercs/ builds up into the high 20s, low 30s... specifically /dark and /FF, and found them to be hands-down the poorest at handling banks. Perhaps at 50 the set starts to shine, but they really didn't have the tools, either in terms of high damage output or interesting effects, that other sets have. So, I put those on the shelf, at least for cranking the difficulty up on banks.
Anyhow, best of luck, if you find any other interesting ways to cope with banks, post. I'd sure be interested
On my Necro/Dark I just set them to follow/aggressive and walk leisurely to the bank racking up minutes, then in the bank follow/ defensive for the LB lobby slaughter then back to /aggressive. But /Dark is great for that
What are some good tricks for dealing with enemies that bypass your henchmen and aggro straight to the MM? For example, on mayhem missions, Longbow usually starts attacking you directly, even if your pets are doing more damage.
All I can think to do is to tell my henchmen to attack one of them, then run away and wait on top of a building or something until they kill their target, then go get them another....... etc. If I try to fight longbow directly, there's just no chance. I'll be dead in a few seconds with all their fire on me.
Sometimes I try bringing a couple of defense inspirations with me, so I can at least last a few seconds. Either that, or maybe that temp power that makes you etherial. Are there any other good ways to solve the problem? Maybe some builds are better for it? I'd like to use something with Thermal maybe.