Mike and Walter final confrontation has been building up throughout the whole series and reaches its climax in the "Say my name" episode you quoted. The two characters never got along from the start. Mike never liked Walt and certainly never trusted him, he also tried more than once to warn Jesse about confiding in Walter.
He has seen right through him from the start and always considered him a dangerous man, so I don't think that he "lets" Walter kill him out of misplaced trust. He simply knows that he's lost everything. He knows he'll be hunted down by the DEA because the attorney he hired to represent his guys has given him up and that he'll be forced to run away, hide somewhere and always look over his shoulder. He's also lost the most precious thing left to him, the relationship with his granddaughter who he was forced to leave behind without even say goodbye. He chose to become Gus's security guy, so to speak, because he wanted to leave enough money to his granddaughter in order to grant her a future. But his efforts were undermined by the DEA who managed to get hold of all his money.
Mike blames Walter for ruining his plan, for killing Gus and wreaking the havoc that destroyed his life's work. I guess that all of this clouded his judgement in the final meeting with Walter and caused him to lower his guard and forget about checking the gun before it's too late.
In the end, sitting on the river bank, he's not even angry at Walter for killing him, he's just fed up. He failed and and he just wants to get it over with (and needs Walter, for once, to "shut the f*** up").