I just want to clarify, when you say first register do you mean the Chalumeau register - from the lowest note of E (3 ledger lines below the staff) to about G in the staff ?
Generally, this can be affected by pad/keywork height
It can also be affected by the mouthpiece itself
I have a variety of mpcs and as I was playing a BC-20 the intonation was quite wild with my Vandoren M13Lyre. But when I use a period correct Buffet/Chedeville mouthpiece the intonation is close to spot on from bottom to top.
Normally I don't use any bumper material thicker than 1/32nds of an inch, such as on the bridge key base (1/64) on the bridge key itself, and throughout the instrument (except tenons which vary). There are exceptions but usually on student level instruments.
Other potential things ... but need more information. (having it in my hands too would be great, but ... )