I'm 99% certain that it's not possible, but just need another confirmation. Is there any way to extend the Translation Manager to work with "3rd party" translation management systems?
Well, the TM does a little more than just throw some details into a package and fire that over to the listening service; consider also that it's updating the TM database, pinging the translator to see if there's anything ready to come back etc.
That said, you have access to the format of the output package and could simple re-format this for the recipient system. You would need to understand what the TM is pinging and what response it expects in the event it's to request the translated packages (they are pull not push) and I guess you could hack it together.
Should you? Well, I don't believe so. The fact you'd need to perform a full test following every product every update or hotfix in the event anything changes (as you're not really using it for the intended purpose) and the fact it's not documented/intended for this.