Barrel Behind the Door

If anyone can help me with this I would be grateful.

Let's say I have a Brain Reasearch Lab on my field with 8 or more counters on it. I set an MST and BBTD and End my turn. My opponent draws. Now here is the deal:
Can I activate MST, destroy my Lab, and when its effect activates chain BBTD?

On a random post I saw on Youtube sbdy sais you can't. So I searched BBTD rulings and saw that it has to be chained on the activation of an effect that inflicts damage. So that excludes cards like Wave-Motion Cannon which you send from the field to the hand to inflict damage (well I don't fully understand that so plz enlighten me on this one). Anyway, when I noticed that Lab's burn effect activates AFTER that card has been removed from the field my hopes started growing. Please help!

Checked BRL's rulings and the first ruling states:

None of the effects of "Brain Research Lab" starts a Chain.

And since BBTD needs something to chain to.... yea.... sorry :P

Barrel behind the door only need an activation of an EFFECT that include a will inflict effect damage (instead of an activation of spell/trap/monster effect's effect). So you can chain it when Wave Motion-Cannon's effect is activated.

It seems counter trap works on a different league.
BRL's effect is similar with Call of the Haunted that will destroy a monster if it's removed from field and you can't negate the destruction with stardust because it doesn't activate..

Since barrel is a trap that reverses dmg it needs something to be chained to, none of BRL's effects starts a chain so he cant use BBTD with BRL

The best you can do is put Prime Material Dragon for your Brain Research Lab trouble. I have 2 in one of my psychic decks, and it works pretty good.