An interesting and useful application. The idea is quite feasible, but the particular models of hardware you linked to are not the best choice in my opinion. The keyfob has addresses that are hard coded either by opening the unit and modifying solder bridges (which is a workshop change only, not a field change), or you must accept a fixed address from the factory. This does not allow much flexibility in the case of multiple units or dogs on one search. The receiver, which would be located on the dog's harness is not packaged for outdoor, on-body use. In addition, it requires a 12V power supply which is not convenient for personal use. It would be much better to find a receiver that is suitably packaged and runs from a couple of AA cells instead.
I will look around and post with better alternatives if I find any. If we can't find a suitable receiver, it might be necessary to design and build something custom, which is not such a large project, especially using modular assemblies.
It will also be necessary to customize a suitable antenna for the dog's harness which is not difficult, but requires a bit of antenna knowledge.