Most probably BB is valuing a straight (Jx) , depends completely on your current table dynamic whether he would fold to a sizeable raise or not especially since you had been pounding on him. People do tend to get stubborn sometimes!

Folding seems to be the right play.

Bet turn, fold river after checking

what do you think BB's preflop range is? I know you said loose but is he any 2 suited cards loose? If yes then fold river as he has plenty of flush combos and you don't make money at 2/5 live poker trying to get people to fold flushes.

If you think BB has a much tighter range preflop + the fact that the A/Q/9/T of diamonds are all out = he has way more QJ/JT combos than flushes(ie54s-87s,KJs). If this is the case I think river bluff will work enough to be +EV

After it goes bet-call-over call on the flop the K is pretty bad card for us so I like the turn check

ive been playing a 1/3 game in niagara that plays like a 2/5 game. I think you played this hand fine. What i find from the older regs is that they wont fold flushes and might even find a hero call with their straight.. they dont mind losing with a good hand and showing all their buddies how unlucky they are lol

i also like the check around on the king which is probably what id lean towards depending on game flow.. however there can be an argument made for a bet.. it is very unlikely for MP to have a better hand than you here because he almost always has to bet for protection on such a draw heavy board and if u know BB well u can maybe eliminate a lot of kings from his range as well .. a bet here while knowing you have the best hand allows u to value draws and check back rivers especially on nasty ones like the 10d....

however id probably still just check turn lol
let me know if that makes sense tho