Am I able to also have a handgun? This will change the equation. With a good sixgun chambered in .357, .41, or .44 Magnum, or .45 Colt, which can reach out and bit and also serve for defensive use, the long gun could be a budget rifle able to shoot cheap .22 LR, or an inexpensive shotgun, possibly even a single-shot.
If there is going to be no handgun in the equation, I would want the rifle to be able to do more with its cartridge, and fire multiple shots fairly rapidly. Keeping in mind the OP is on a budget, I would look into a pre-owned centerfire lever-action rifle, chambered for a magnum handgun cartridge, or the good ol' .30-30.
For the purpose of this exercise, I factored out my job situation, as I must buy my own duty firearms. The one long gun I virtually NEED for the job is a Remington 870 shotgun. (The other PD-authorized choices are the Benelli M1 or M2.) Of course, an 870 would not be a bad candidate for the one long gun. I think, however, I would rather my only long gun be a rifle.