Delay in picking gear is def a gear / gear oil issue, but jerking while engaging in reverse could be gear or the engine/gear mountings being damaged. So while eliminating one thing at a time, you will deal with actual root cause of the problem and spend less time and money.
He has already replied you about the repair of gear is rebuilding only, you cannot fix anything inside without removing the gear box completely outside the car. Replacing partial parts is not advisable if you plan to keep the car for long, as other part will wear and again after a year or two you have to open the gearbox. So complete rebuild with OEM quality component is the only way out.
So to bite this bullet or not, you need to dig the history and ask why they have not changed the gear oil ever (in case).
If you have been maintaining with dealer then ask them for all the records showing full details and see when they have changed the gear oil last and what is the recommended kms for gear oil change in your owners manual? Does it match correct....?