The iphone/ipad video with mic should work just fine in most cases. The cameras on devices have improved so much in the past few years that a different camera really isn’t necessary…especially with a good external mic.
Will it be just you and a pianist or are you recording jazz with a rhythm section? We had studio recordings made, but only because we had some contacts. One excerpt that didn’t require a rhythm section got re-recorded last minute on an iphone with no external mic and it didn’t seem to matter.
All the schools S applied to accepted a wide range of video and audio formats. Some had size restrictions. You can use imovie or whatever video editing software you have to trim, size and format for download if needed.