Your best option is to hire someone, and they should have the gear as a part of their package. Since it's only for a couple of hours it should be well within your $300 audio budget. Google up "sound mixer" "production sound", and "remote recording" for a local contractor.
An optimum set-up would record the piano in stereo (two [2] channels) and the vocalist (one [1] channel), a total of three (3) channels. The kit would include a stereo mic or a matched stereo pair to record the piano, a single mic for the vocalist and a four channel audio recorder/mixer set-up. If mono is okay a mic on the piano and a mic on the vocalist will work fine.
You could rent what you need. The one day rental should be relatively inexpensive. Contact a local rental vendor and check out their rates. 4-track digital recorders usually go for about $100 day, mics anywhere from $25 to $50 a day. As a package it would probably be $150 including the accessories (stands, cables, etc), probably a lot less. You should do a sound check prior to the performance to get the levels; keep in mind that the performance is almost always louder than the sound check.
$300 will not get you very much in the way of audio gear. Reliable, durable field recorders with good sound start at about $500. A passable shotgun mic with a boompole, shockmount and cables will run at least $300. (A shotgun mic is not what you need for this particular gig.)
Just keep in mind that you are filming a SINGER and the accompaniment, which makes this primarily an audio gig; the visuals are secondary. If it doesn't sound great no one will care how beautiful your shots are.