That's actually how I started out.. I got clients before I had my business open or even thought about it. lol
Yes it can be done, but it's tedious and time consuming. As for the type of billing, if you bill for nephrology, you can bill for mental health.. just a lot of authorization, auth tracking (hence the tediousness of not having a good pm software) I'd also have to ask myself why she doesn't want to invest in a software.?
Also another suggestion, I know you said you are not ready to start your own business..but regardless you SHOULD setup a contract with her and work as a sub-contractor at least to avoid any problems. If you set yourself up as an employee, she will lose out on all the benefits of outsourcing yet if you work as a sub-contractor you both get the benefits, you have a contract and your both covered.