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.