I asked the same question 2 years ago for my son who went through the application process last year. We ended up going through the professional studio route for the audio portion because he had to record with a whole jazz rhythm section, i.e. three other instruments beside himself, and also one of the pre-screens was audio-only.
Here’s the thread if you’re interested.
http://talk.collegeconfidential.com/music-major/1777432-quality-of-pre-screen-recordings-p1.html
We took our own video camera, a Zoom Q8, to the studio session. My son integrated the audio, which the studio provided on a CD, and the video. This took some effort. I think he used Adobe Studio, which he already had. He passed all of the prescreens, including Juilliard. In hindsight, we both think he would have passed all of his video prescreens with just the Zoom Q8 recording. The Q8’s strength is in the audio more than video and takes external mics.
We hired a pro drummer and bassist; pianist was a combo-mate. I think having a pro rhythm section (so the accompanist in your case) mattered much more than the studio environment. He had to record 9-10 songs, so not having to rehearse or compensate for the rhythm section had a big impact.