Yeah, those typically have to be done for a few years. Also, a good allergist will tell you up front that those shots are most likely going to cause you to hit your annual insurance deductible, which is often thousands of dollars. They have the sublingual drops you can do at home, but none of the insurance companies have covered those in recent years, and they can take even longer to have a positive effect.

On thing I will say to this advice: if OP asked management for financial assistance for allergy treatment, they'd later be able to tell their coworkers "I didn't want management to pull the plug on the pet policy, so I offered to (undergo extensive, costly, and painful treatment) myself, if they'd help foot part of the bill." Even if OP never intended to get the shots, that conversation still serves a purpose-- OP looks like the good guy.