G. Livermore,
It sounds like you are describing a Beam to Column field moment connection where the beam is framing to the web of the column. Texas "T" connections sometimes do have stiffeners in the web of the column that are thicker than the flanges of the beam(s) framing in. Most often it is because of 2 different depth beams framing opposite the web and the thicker stiffeners allow aligning the bottom flanges.
Is it a requirement per the specs to remove the backing strip or can it remain in place?
The AISC has a publication "Steel Design Guide Series #13" (copyright 1999) that addresses this issue and I'll quote so I won't mess it up,
" Appendix "D" (starts on page 95) Titled "Column Stiffening Considerations for Weak Axis Moment Connections" Note #3 at the top left of page 99 states"Connection plates that are made thicker or wider for control tolerances, tensile stress, and shear lag need not to be ground or cut to a transition thickness or width to match the beam flange to which they connect." Figure #1 is along side this paragraph and shows the thicker stiffeners with out any transition of the thicker stiffener plates.
Hope this answers your question about transitioning the plates,
John Wright