A 3/4" particleboard underlayment for carpets would be your lowest cost alternative. Combined with the conventional tackstrips, padding and carpet installtion, the tile and carpets should like up.
As you approach finishing on a new home, the budgets get pretty tight. I think you need to meet you builder halfway on this an pay for the materials. 3/4 particleboard is fairly inexpensive (about .22/foot). The floors will end up even and much quieter. The only problem at this point is making all the doors open properly, and hopefully you don't need to do it in the kitchen where cabinet height would be affected.
My impression is that the builder should have anticipated the situation because 1-1/4" is a trip hazard any way you look at it. Its not a step or split level, its just a minor elevation change waiting to grab you.