Currently, pathogens "use" certain resources in a host, and then the host's immune system creates antibodies that eventually kill the pathogens (or the pathogen kills the host).

This is a pretty good understanding. However, antibodies are only one tool in the immune systems of complex animals. There are also a number of specialized immune cells that perform jobs like patrolling the body to try to detect foreign cells and objects to "label" as invaders, or phagocytes that try to envelope and digest foreign cells and objects. Additionally, there are strategies like making the body a less hospitable environment by increasing the temperature (fever). Antimicrobial chemical production is also very common in single-celled organisms but identified in fewer complex animals.

My question is: mirror or otherwise, could a pathogen "hijack" something other than usual as a resource?

Let's say, I don't know, Prime Pathogen A normally uses Prime Protein A, Mirror Pathogen A would require Mirror Pathogen A. Is it possible for a host to have a Prime Protein B that meets Mirror Pathogen A's requirement–perhaps not perfectly, but "good enough" to sustain Mirror Pathogen A?

Possible but, when looking specifically at the hypothetical "mirror" biology, less likely and probably would require specialized adaptations. The main problem that they would need to overcome, which is rather uncommon in known life is the mirrored shape of important biological molecules. Specifically, amino acids and sugars.

Both of these types molecules have a property known as "chirality", where the atoms in the molecule can be oriented in two ways, while still all being bonded to the same neighbors. These are known as enantiomers and designated with the prefixes "levo" ("l-"), meaning left, and "dextro" ("d-"), meaning right. Biology as we know it is very specialized to make use of l-amino acids and d-sugars. There are some exceptions but they tend to be rather specialized and less energetically favorable.

Some organisms can break down d-amino acids and use their atoms as nitrogen sources to build new l-amino acids. It is much, much more common, however, to use l-amino acids from either the host cells or their proteins. This takes the pathogen far less energy to accomplish as well as not requiring evolving specialized enzymes that most organisms don't have.

So, in the case of the mirror pathogen, it would have to have the ability and dedicate the energy to breakdown l-amino acids to build its d-amino acids. This would, indeed, likely result in slower growth rate. That, in itself, has a lot of potential to limit pathogenicity as high growth rate is generally required to be problematic. Phagocytes will still try to envelope and remove foreign objects that they don't "think" are alive, so, the mirror pathogen could face risk of excretion if it is not able to reproduce quickly enough to maintain a population.

So, overall, yes, hypothetically possible and even plausible (if the mirror organisms are able to metabolize d-sugars and l-amino acids) but less likely. Pathogenicity requires the ability to grow a population fast enough or otherwise evade the immune system which is also able act on "invaders" that it does not recognize as biological (maybe being really good at building a bioplaque that makes it difficult to move or use of toxins). Simply having mirrored chemistry alone is probably insufficient to evade the immune system enough to be pathogenic.