I have opened dialogue with him before and he said he would tell the admission committee about our contact. I have not been rejected yet, but I would really like to work with him (payment does not matter). I am already enrolled in another grad program without many resources, so being able to contribute to his project would be a blessing to my resume.
No, it isn't inappropriate to ask, but don't expect that much can be done. In the US, grad admissions are normally up to a committee that evaluates your entire application. The professor may have something to say in your favor, but, unlike other places, it may not carry much weight.
But, if you mean whether you can ask the committee (rather than the professor), then it probably won't be considered at all (most places) as it is beyond the application materials. Rules may prevent such additional "information" from being considered. There are exceptions, as every department is different from every other in the US.
But, even if you are rejected, you can still work with him externally if your own advisor agrees. That is a matter of personality and such. So, you also need to ask the advisor of the program you are enrolled in if you need to continue to work with them. And note that there is little advantage to the professor for working with an external student unless your research ideas are especially interesting. There wouldn't be any pay increase or load reductions for such things.
But, you can ask.
You can ask but only after you get rejection.
You must have some solid grounds like some work in that field (lab work, research work, training, internship or something that will really impress them).
My son applied for MS in a University and was rejected. Then he asked the professor or committee. He told about his achievements and other work. They agreed to take an interview and finally he got admission.
But all should be done very politely.
You're perfectly allowed to work for anybody you want who is willing to employ you, under any conditions that don't fall outside of applicable labor laws. This has nothing to do with any acceptance or rejection from an academic program of study.
If you see some advantage of taking such a position, there is no reason not to. It is completely "appropriate". There may be fewer options for pools of money the PI can pay you from, but your question states that being paid is not an issue. If you are not an "official" employee of the university, you have no formal protection that traditional workers do, such as workman's comp if you get hurt while working, unemployment insurance, health care policies, ...
Beyond that, only you're in the position to know whether such a role will serve to advance your career. If you plan to use the experience to strengthen future applications to this program or others, so you get accepted, it is possible that it is advantageous. Beyond that, though, my own opinion is that there may well be better ways to spend your time.