There are several possibilities, and we probably won't hit them all, and the probability of any
one of them being *your* problem is about 5%.
Is it a pump pressure problem that is causing the delay in proper pressure? Does the filter need replacing? There is a very commonly used transmission filter on many Chryslers that is almost identical to our filter. It isn't an exact fit into the recess cavity in the transmission, but it's so close, you can push it in without much trouble,
but the nipple that points upward and plugs into the pump inlet is a bit shorter than the nipple on ours, so the pump sucks air. The parts guys are so used to selling the other filter that they can get in the habit of selling them to everyone without looking it up. When I got my first Concorde, someone on a Chrysler newsgroup warned people of the wrong-filter problem, and I kind of forgot about it. Darn if a few months later I did a transmission fluid and filter change and turns out the mail order dealer that I bought the filter from shipped that other filter. I only noticed it because it didn't fit perfectly into the cavity, then I remembered what the newsgroup guy posted, so I pulled it out and compared it to the old filter, and the nipple was noticeably shorter. Just glad I noticed and remembered before putting the pan back on and refilling.
That's just an example of what I'm talking about with several possibilities, but the chance of any one of them being *your* problem is pretty small. But do think about if your filter might be slightly clogged or having been replaced with the wrong one.
Then there are the wiring and range sensor possibilities already mentioned, except it seems that your problem repeats itself at each startup and goes away.
Back to the TCM being the same part number or not, I can tell you that my '98 TCM went bad, and after researching it, I bought a used 2000 one, only paying attention to which engine it had been used with because the gearing is different for the 2.7 vs. 3.2/3.5. Worked flawlessly, and still is. You can't expect the mechanics to know every line of car and all their quirks and variations like DIY'ers who own a particular car for years and hang out on forums for years like us (IOW, a pretty sad bunch of people).
But the mechanics pretty much have to tell you to use the same part to be on the safe side (same with the salvage yard interchange lists - they paly it safe and often don't list an interchange when the parts are in fact interchangeable).