Granted there is very little any of us actually know about your scenario but from what you are describing, i think you have the roles of the players a little backwards. If SPS is paying last years taxes, chances are they are the servicer. That would imply they will also pay this year's taxes which would eliminate you being able to "Buy" it. You wouldn't be able to buy it regardless unless two situations exist. 1) you contact the owner and they agree to sell it to you and you purchase it or....2) the property goes into foreclosure and subsequent foreclosure sale and you show up on the courthouse steps and purchase it for cash.
Assuming what i've said is true it implies a couple of scenarios. If SPS is advancing for taxes and they are NOT in a workout with the borrower, chances are they will foreclose eventually if they have not already.
SPS doesn't "Purchase loans from banks when they fall behind on payments". SPS is a subservicer. They are contracted to service loans on behalf of banks and other owners in exchange for a fee to do so. SPS doesn't force any homeowner to foreclosure by raising rates and monthly payments. That's just not a fact for any lender or servicer in business today.
Homeowners today are "forced" into foreclosure when they don't make their payments or when the violate other covenants of their loan (Illegal transfer, failure to pay taxes, failure to maintain insurance, failure to maintain the home, operating a meth lab, etc..) Homeowner's "Payments are raised" or their "rates are raised" when they have an ARM loan that adjusts upward or their insurance increases or their taxes increase. Lenders don't just raise payments or rates. Some loans do have default rate provisions that kick in in the event of default but not any consumer loans that i'm aware of.
...so based on what you wrote, there isn't any process to work here. Buy it at foreclosure sale if it goes to sale or buy it from the owner if they sell it to you. In either scenario, SPS is not a factor in the equation.
There is one final scenario - you could buy the tax lien at a tax sale but again, if the servicer is advancing taxes, there is nothing for you to buy.