Idk if it works on Bee, but as I used on my previous lineup Mabui/Tenten/Darui, I controlled who will be the target of Mabui/Azure standard heal when all of them have full HP, so I cn always cast her mystery despite prompt acupuntures/immobiles.
What I did was to "place" Tenten first into the lineup, before all other ninjas including the main. Now, you might wanna say "the main ninja cannot be removed from the lineup".
Yes, you cannot, unless it's Matsuri/Survival. Now, to make Tenten the target of standard heals, drag her up, and put her down back to her tile. Proceed to the next ninja
in the priority order, drag and drop on same tile. The last ones you will be dragging up and placing down would be the healers, or in your case, Bee.
I personally haven't tried this on Bee/Kurenai, and other ninjas that target allies. You can try.
As for the Mabui/Tenten/Mei or Darui team, that strategy works 100%. Certainly comes in handy in space time autos.

In short: The target of standard heals is the ninja that is "first" to be dragged up and placed down on its tile. Make sure you go all the way through the fourth ninja to make the priority order work. The fourth ninja to be dragged and dropped down is considered the "last" ninja, and thus, will not be the subject of standard healing unless its hp is the lowest (with suffered dmg).

Fun fact: The "ninja with the lowest life" description in most healing abilities is based on the targets % HP in proportion to its own total HP pool. Thus, if all your ninjas have full HP, the one with the lowest HP value don't necessarily get to be target of the healing standard. The system seems to calculate HP proportions relative to its full value, based on percentages.

Yes, that's why I said I have not tried it with Bee/Kurenai or other ninja with same manual-target supportive mysteries.
Just thought it could work the same as I did to the Mabui/Tenten team. Sad it doesn't work.
Now, some people say Bee's Yo! actually targets ninja with the lowest HP pool, so try to arrange your life magatamas/runes so that your target ninja gets the lowest of the 4 ninjas. Careful not to overdo it, as your target ninja's survival may be compromised.

You basically just rephrased what I said. :D
I thought he tried it but didn't work, so I couldn't think of any other ways to make it work.
Hope it does work for him.

That wasn't the case for me. I've had the same lineup, same positions for those ninjas since I gathered them up. I wasn't "moving" them anywhere. I merely lifted one ninja up, and placed them back to their own tile, following the order of making Tenten first, then Darui, Azure and Mabui.
However, it gets messed up when I change tabs, so I had to redo the order again when I switch back to that tab.

True. :D

Yes, I understand how moving ninjas to switch tiles with other ninjas could mess up their gears.
However, I said I never really move any ninja elsewhere.
I only lift one ninja up and place it down to its own tile. That way, that ninja still wears the same gear.
What me and others have stated here was: It's pointless to remove all ninjas from the lineup when you could achieve the same result by just lifting ninjas from their tile, and placing it back. That way, they wear same gears, and saves the trouble of messing their gears had we removed them from the lineup.

You know why it doesn't work? Coz you were only doing it on those 3 ninjas.
Just do this step-by-step:
Arrange your ninjas to your ideal positions.
Now do this in order,
1. Lift either Roshi or Minato (whoever is your 1st move ninja), then place it back to its tile.
2. Lift Bee off its tile, place it back down.
3. Lift the 3rd ninja, place it back down.
4. Lift your main ninja, place it back down.

DO NOT remove them from the lineup. It will mess their gears up. Just do the ones I mentioned, lift a ninja up, then place it down back to its own tile.
I just tested it on Kurenai, now she always shields my Roshi first.

CAUTION: When you switch tabs, the order will be reset. So when you go back to your main lineup, redo those steps again. Switching their gears also reset the process.
Changing chakra runes will also reset it. Basically anything that alters their stats will mess the order up. Just keep doing it over again if so.

Hope this helps.

Edit: If it still doesn't work, then the problem here is miscommunication. I certainly tested it and it worked in all the times I tried. Please re-read the steps carefully.

You're talking about prompt orders relative to positions/move order, as in 1st, 2nd and so on.
Yes, runes will change if you switch the order of who goes 1st and such.
I wasn't talking about that either, and it's not even a matter of prompt order (the steps I mentioned), just something to change the target of a support mystery
to whoever you want it cast to. So, when I said "order", it meant the order in which we place ninjas into the lineup, not their actual move order.