That mileage sounds perfect for a shimano slx chain. It all depends untill what level of wear you ride the chain. The 800km might be on the high side of % of wear. The SLX is their second from the bottom spec level, so do not expect to get 1500 or 2000km. If you want the longest lasting chain for your Shimano drivetrain, go with xtr
Your gears jumping could also be due to your current derailleur that is worn out. Just ask the bikeshop to temporarily replace the derailleur with a new or demo shimano one, and then you can test. Shimano derailleur are not as expensive when compared to sram.
Another thing that tells me that your derailleur might be the issue here is the shop that replaced your pulley wheels. Slx and deore pulley wheels do not have bearings. Instead they use bushings and have a lot of side to side play, even from new. The shop most probably replaced the drivetrain and when testing they weren't happy with the shifting. So they checked the pulley wheels and saw that play and replaced them with csixx pulley wheels, with bearings inside of them. Most of the time it improves the shifting slightly but never perfectly.
If they did the whole drivetrain replacement I hope they at least changed your gear cable and cable housing
