I think I would be fine. I already have limited wealth and I'm directing a significant portion of it to the betterment of the world around me. If my wealth/income increased tenfold (as an example) I don't think I'd be able to find a use for more than, idk, 15-20% of the additional money? So I'd just heavily ramp up the funds going to charities and local causes.
Extending this out to the infinite wealth scenario, I'd be able to help more causes so I'd need to find out which charities are actually putting money to use for what they say they are, extend my mindset beyond local causes and to the whole globe. Almost certainly drop some fat stacks into buying out basically every fossil fuel corp and start the process of getting that shit phased out. Sure I make some terrible people rich(er) in the short term with the buyouts but fuck it, the environment needs to be prioritized.
How I'd convince this hypothetical entity? Idk, probably just explain all of the above and I guess show some hard data on my track record for this. If the entity is determined to put the money in the hands of someone who is going to do good, either I'll be good enough or I won't.
If its not infinite and "just" billions then I'm probably going to funnel about half of it into local (national) causes and the other half into affecting what positive change I can worldwide. Process for convincing the entity remains the same.