Sounds to me like this is a tiptronic and there is a code set in the TCU (could be literally anything - perhaps not even related to the reversing shenanigans). With said code set, limp mode is activated which gives you exactly 1 forward gear: 4th. The car will be dog slow (but driveable) because it's starting out in 4th gear.
I'll bet the shop you took it to did not even check the codes in the TCU (or tried to do so with a generic scan tool and were unable) and therefore have absolutely no idea what is wrong. Rather than diagnose, they are merely guessing it is some physical problem with the transmission which will, of course, be fixed by swapping out the entire transmission for $1800. If we are using their logic, if you have a problem with the car, you might as well just replace the entire car rather than figure out what's actually wrong with it. Because that's just simpler.
I would highly, highly suggest you get the actual codes read on the TCU. Get them printed out or written down verbatim. Then clear the codes. Assuming the fault was not ongoing, once they are cleared, the car should drive normally again. If the code will not clear (or comes back after a short drive or 3), then there is an ongoing fault that will need to fixed. And now you have the code which should give you some information about what is wrong.
You've provided very little information here. We're not even sure if you have a manual or automatic transmission. So the above is just my best guess for now. But, if it's an automatic, get the codes. And don't believe any transmission diagnosis if they can't provide the stored codes from the TCU. The codes are easy to get. You can retrieve them with an XP laptop, a $10 ebay cable and free software. So don't take any crap.
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