P1776 is the code for Solenoid Switch Valve latched in low/reverse position. The solenoid switch valve is located in the valve body itself, not in the solenoid pack.
The proper repair is to diagnose to see if it needs a shift cable adjustment, shift cable replacement, valve body cleaning, or a new valve body.
They need to check the TRS portion (Transmission Range Sensor, tells the computer what gear you have it shifted in) of the event data to see if it reads TR2 or OD. OD means it needs a valve body cleaning, or new valve body, the TR2 means they need to check the cable for binding, proper adjustment, and proper movement.
The total cost parts and labor should be around $550 for a replacement valve body. That would be the most expensive of the repairs listed above.
Most shops actually mistakenly think the code refers to the solenoid pack.
Best thing for Chrysler transmissions is to take them to the dealer when you have a problem with them. Chrysler transmissions are electronically controlled, and have many different quirks to them that many independents don't know about. The dealer trans techs know them the best, and have the best information on them.
Second choice would be a WELL qualified independent shop.
Last choice is a trans specialty shop. They're notorious for selling replacement transmissions when they really only need a minor repair. Sometimes it's due to ignorance and/or just not understanding how a Chrysler trans works, sometimes it's due to increasing profits as much as possible. Not all are like that, but enough that I recommend staying away from them for diagnostics.
They may have not recognized the P1776 code since it isn't listed in most diagnostic code lists, and just assumed it meant it needed a transmission replacement.
Basically, what happens, is the valve itself is either hanging up, keeping it from shifting properly, or the cable isn't allowing the proper movement for the TRS to know for sure which gear it's in, it gets kinda in between, so it won't shift properly.
BTW, The fact that it shifts and runs fine when the check engine light is not on is a clear indicator that it does not need a new transmission.