Though I cannot make a specific recommendation for your area, I can offer some thoughts, as I have redone many interiors, and will be recovering two seats myself next week.
Agree beforehand with your mechanic as to what standards they would like to see for the materials and workmanship. Depending upon the certification basis of the aircraft, there could be an expectation that "parts" be used rather than "material". This is probably impossible, so you'll be using material, just agree beforehand as to what, and how it will be sourced/documented. The material must be at the very least "flame resistant", though more rigorous standards could also apply. Make sure that the material is chosen to comply, and this compliance can be authoritatively documented if asked. In the most extreme case (and I have encountered it) a covered seat cushion must comply with flammability requirements as a composite assembly, rather than the covering material alone. This complicates matters, avoid if possible. For light aircraft it is probably possible to interpret away from this with keen thinking. If you can reuse the foam part of the cushion, that's a good way to not have to demonstrate flammability compliance for the foam - if someone is asking.
Unless you're recovering to simply sell the plane, for the material - choose a quality leather. The labour cost to recover the seats will be the same regardless of the material you choose. You may as well have seat covering which is tough, nice to sit on, will last for decades, will not fade, and can be cleaned of nearly anything (barf). And, virtually all leather passes all flammability requirements (though some leather may not be documented as such). Leather is much nicer to work with, so the quality of the job will seem better for the same effort, and last longer looking good. I did my C 150 seats in leather nearly 30 years ago, and they still look presentable. I have never redone seats which were not done in leather. One plane I bought, had just had the seats done in very expensive, FAR Part 25 burn compliant (airliner) seat cloth. Within a few years, they were faded and looked poor, a year later the top corners started to deteriorate because of UV. Then I spilled some hydraulic fluid while topping off the brakes, and that was that. I recovered them in leather, and they've been fine since. This aircraft type tends to have people standing on the seat cushion to enter and exit, so the tough leather handles that much better.
Just for reference, here in Canada, the owner/pilot of a certified privately operated GA aircraft can perform "elementary work" themselves, and sign the maintenance release for that work. Included in that list is: " (14) repair of upholstery, trim and cabin furnishings; ". Okay, "repair" and "recover" may be interpreted differently, but it may be worth a look to your local regulations for a similarity.
It may not need to be an "aircraft" shop who does seat recovering, automotive, marine, or skilled furniture upholstery shops can probably do fine work, ask to see samples.
When recovering the seat backs, add pockets - lots of pockets! Pockets can be sewn up out of fabric to save cost. I like red fabric, which you just see a hint of at the top. Save all the off cuts, you might want to accent with other pockets or patches of the same material on the sidewalls.