Ditch the carrot and see if you can locate a farm or feed store that sells horse hay by the bale (not cow, usually poor quality), and if you have the room, purchase a bale, or ask if you can purchase half or a quarter of a bale if you don't have a lot of room. Ensure that if your rabbit is over 6months in age, you use Timothy or Orchard, Botanicle, Meadow, Oat hay, etc... NOT Alfalfa/Lucerne hay as this is too rich in protein/calcium.
I purchased an 80lb bale of Orchard grass 2 summers ago for my 4-5lb rabbits. They are still working through it. That $10 went a LONG ways.
You can supplement with pellets, 1/4 - 1/2 cup per 5lbs or rabbit, or so, per day, so you don't need much. Would preferably be timothy based lower protein pellets but so long as the hay IS NOT alfalfa, some alfalfa based pellets wouldn't be the end of the world as these are usually easier to find. But the bulk of the diet should be hay, and since that's really inexpensive, youre not spending a whole bunch of pellets.
You can also slowly introduce leafy green vegetables and herbs.
Maybe some locals can pinpoint good stores in the OP's area?