I'm MtF, but I gotta say....it sounds like you're getting a pretty 'authentic' male experience <3
What you're feeling is perfectly natural, it's what testosterone does. Just because you're receiving testosterone from an external source instead of producing your own doesn't make the experience any less natural. You're discovering a common (frustrating) experience that most cis men understand well. This means your HRT is successful!
If it's becoming a serious problem, you may want to talk with your endo about backing off the T a bit. That should at least help reduce your drive.
It's very difficult and frustrating having a sex drive that's higher than your partner's, I can sympathize completely. I wish I had some better advice, but I've had to learn to just deal with it and enjoy things when they happen. "Once a day" would be a dream come true, but that level of desire seems to be uncommon without male levels of T, so we just have to learn to go at our partner's pace. Honestly, one of the things I'm looking forward to about my upcoming MtF HRT is the reduced libido. I think you can understand why -- sex is great, but when the drive gets so intense that it becomes a need many aspects of life start to suffer, particularly in the relationship department.
My absolute ignorance of FtM HRT effects may be showing here, but would something like a Feeldoe help you out? It's basically a strapless strap-on that is inserted vaginally. It may be too uncomfortable under HRT, I'm not sure. But it might allow you and your girlfriend to enjoy some intimacy and let you have the experience of penetrating her while still receiving pleasure yourself. Regular strap-ons are kinda lame from what I've been told -- it's difficult for the wearer to enjoy them. But the feeldoe (or enjoyus, or other similar products) seem to actually provide a more immersive experience for both wearer/receiver.
It sounds like there are some things you and your girlfriend need to talk about and work through regarding your sex life. If you feel like she's just in it for an ego boost, that's a devastating feeling for you that really can't be ignored. A relationship councilor may help? Obviously this should be brought up gently, and be conscious of the T while you do it -- it's so easy to get frustrated and lash out when sexual tensions peak and testosterone is raging. Learning to manage that is an important part of being male, and it's not always easy.
Best of luck to you, keep trying! You'll find something that will work eventually!