The Weber you would need for the 1.3 is a DMTR.
If the Pierburg you have is the original 2e3 (curvy full-load enrichment pipe going into top of secondary barrel?) then it has no waxstat, only the 2e2s (with straight full load enrichment pipe) fitted to 1.6/1.8 engines have this feckin thing.
Instead, the 2e3 has a stepped cam arrangement which governs the primary throttle opening/idle speed during warm-up. The mechanism is intricate, including a tiddly spring or two It may just be dirty and gummed up, it may be a little broken. Give the mechanism a good hosing with carb cleaner to clean it all up and see if things improve. You'll find it in between the choke water housing with the two coolant hoses, and the main body of the carb.