Verified answer

not 100% sure but to me it would mean something like, once that that company president decides to do something once, no matter what happens, he won't quit or anything.

～たりする means to do ～ as well as other things, do things like ～. You can combine clauses as well: テレビを見たり本を読んだりする means doing things like watching TV and reading books (not simultaneously).

I'm not aware of ～たりしない having any other nuances than just meaning that other acts can happen too.

~tari suru is an ending or construction that means a person does one thing and alternately another.

tabetari hanashitari shimasu for example means to alternately eat and talk. I can understand the first part of the sentence as ano shachou wa ichi do yaru to kimetara once the boss has decided to do something. And, then the rest too like nani ga atte mo yametari shinai. I see your problem. I have never come across that construction as well. But it seems pretty obvious from the context what is meant that whatever happens he won't give up something he has started.

Perhaps since no alternative is mentioned yametari ONLY shinai. He doesn't consider quiting. ha ha ha this is much easier when it is not something that I am translating myself.

I can not stop the President decided that once and do it "or not", no matter what.

write it after "once and do it"