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Why don't you go to your your local dealership, and ask a service writer what upholstery shop they use for tops, repairs and replacement. You do know they don't do it there, they farm it out, to local shops, then go ask questions, check the place for cleanliness, the turn around time is usually 24 hours.
I also would replace the headliner when you have it done.
I just had a 92 done this spring. Labor was $900 at an Upholestry shop. I bought the top for $425, shipped to me. The top was a Robbins Brand Black Canvas one with a plastic window. I already had a convertible weather stripping set. It was another $260.
Looking to do this next spring on my 87 so any watch outs, tips or "why I am doing this I might as well do ..." would be great.
When I had the top replaced on my 89 I had the shop do a glass window conversion. It does take up a little more room than the plastic window and requires a retrofit kit for older models, but I thought it was well worth it. IIRC everything cost me ~$1,600.
with me supplying the weatherstrip and glass window conversion kit, i paid about $1200
i had bought the hard to find discontinued kit to convert to a glass rear window like the 94-96 had.
i believe you could convert to a glass window without the kit, but the kit is desired bc it provides pads and stops for the rear glass window to ride on.
the installer noted my headliner was in good condition so he flipped over the headliner bc the other side wasnt faded.
the installer had a humble small shop in rootstown, ohio. He did a good job with the install and the rear glass window conversion kit (it didnt have good instructions)
what he did a bad job at was the weatherstrip installation.
i recommend doing this yourself. It can be done separate from the top install.
he did NOT do a good job removing /scraping the old weatherstrip glue out. Even though I told him too.
he also stripped and lost a couple screws from the weatherstrip retaining channels
frankly, no one cares about your car more than you do, so do the weatherstrip part yourself. Like I said, you gain nothing having it done at the same time.