I agree that you should weld in the anealed condition first. Your 1100 deg temperature is wrong for normalizing 4130, and the rate should not be 1100 deg f per hour per inch. 100 deg per hour per inch would be more like it. Parts should be stress releived also. D1.1 does not cover 4130 material, although I once had a 4340 job that was to be done in accordance with the welding rules in D1.1, other than the material rules. Procedure qualification was performed using D2.2 as the guideline.
As far as welding, performing the weld on material in the annealed condition vs. normalized would not be a problem. The initial heat treatment condition of the base metal when performing the weld is not a variable (essential or nonessential) of ASME IX. But, if you want to normalize after welding instead of performing the 1100 F stress relief, then that change would not be permitted without requalifying the WPS. Normalizing temperature is above the lower critical temperature and the 1100 F stress relief is below the lower critical temperature. This change would require requalification by QW-407.1. Metallurgically this would have been acceptable, but as far as complying with ASME IX would not be acceptable. So, the QA folks were basically correct in not allowing the work to proceed in a manner different than specified in the WPS.
Technically, the materials should be qualified in the heat treat condition that will be seen in production. Welding on annealed 4130 and normalizing afterward is, in no way, representative of what will be seen in production. The recommended normalizing temperature for 4130 is 1600F, followed by air cool. This is a true "heat treatment", causing microstructural change both upon heating/holding at 1600F and air cooling back to room temperature. PWHT of a weldment in 4130 is done at a temperature below that which will cause microstructural changes.
Something, not mentioned, is the possible adverse affects on the weld metal selected, during the normalizing treatment.
In general, you can do almost anything in weld procedure qualification that you want, but the resulting WPS must then be written to duplicate, in production, exactly what was done in qualification, e.g., to do what you propose would require welding on annealed materials and then normalizing the entire weldment. Not very practical, especially from a size, potential distortion viewpoint.
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