The stud problem you have is caused from the nuts not being tight.
Make sure you are using M8 nyloc nuts and m8 wave washers in the hub studs.
Make sure you have decen hub studs in the hub that are not made of cheese like some cheap East Asian reproduction studs that have been know to shear off the first time you tighten the wheel rim.
You donnit need to open the engine to replace the rear hub oil seal. Hlremive the hub and prolerly clean all that area with acetone or brake cleaner. Change the seal. Ensure that there are no cracks around the area where the bearing seats in the engine cases. Things need t be very clean to see things properly. Inspect very carefully. Some reproduction rear hub bearings leak around the outside of the bearing race between the case and the bearing. Sometimes this can be stopped by using loc tire sealant when you install the bearing.
There have been bad batches of rear hub bearings that simply don’t fit the casss properly and they are paper weights.
First off: remove the hub and investigate the source of the oil. Inspect the splines in the hub core and if damage replace the hub and the cone that it seats on
Secnd: clean the hub and replace the oil seal and the brake shoes. If thebshies have seen oil they are contaminated and need to be replaced. People use brake cleaner and say that shoes are reusable. These are your brakes, replace the shoes.
Third: replace the hub studs, you will need a 12mm socket probably ground down flat on the chamfered edge inorder to grab the stud properly. They are reversed and thread in backwards. https://scooterfix.ca/products/lambretta-hub-stud-19044002-each
Lastly replace the rim and get new nyloc nuts and washers on that rim and the hub. replace all 8 of each. While the rim is off of the tire consider changing the inner tube if its older, shows rust from the rim, or is pinched in any way shape or form.
Figure out what’s causing the problems before throwing money at expensive parts that may not solve anything at all.