Does having an approved patent increases the chances of getting monetary compensation (settlement?) related to infringement claims? How common it is (if at all) for a lawyer to accept a case where the fees are charged only if monetary damage is obtained?
Having a granted patent is indispensible to getting paid for patent infringement. Without a granted patent, you have no enforceable rights. Then the granted claims must cover what the allegedly infringing software is or does. Do you know enough about both of, what claims you might get, and what the "similar" software actually is, to have a reasonable basis for thinking you could have a patent that they infringe? (Just a question for you to think about; not asking you to provide any detail on this forum.)
I believe there are regular law firms that might take an infringement case on contingency, but they might be hard to find. On the other hand, there are companies such as patent holding companies that will take cases on a contingent fee basis and/or simply purchase a patent from an individual inventor, and then pursue the infringer in their own name. The only one I can name off the top of my head is Rembrandt Technologies, which as far as I know pretty much has the sole business model of suing companies for infringement of patents that they (Rembrandt) had acquired from other parties.