There are some broader issues to consider (described in other answers and comments), but a couple of ideas to address the specific question as asked:
1. Request to not be on speaker phone (and don't mention the girlfriend)
I've heard versions of this request before. Speakers and microphones vary in quality, and it's not unusual for the speakerphone to be less than ideal. Maybe it's hard for you to hear the other person with the microphone so far away from their mouth and ambient noises in between, or maybe there is an odd echo that you find irritating or distracting.
You can gesture towards an issue like this and ask (or at least express a strong preference) for not being put on the speaker when you talk. Not everyone will accommodate you, and not in every situation (I know my mom dislikes when I use the speakerphone, but if I'm cooking and she wants to talk right then that's her only option).
2. Invite your friend to events with no option to bring another person
If you buy two tickets to a show that you think will sell out, and then invite your friend once it has sold out, there's no opportunity for him to bring her along. There are plenty of situations where it is plausible that tickets could be hard to come by or available in only limited quantities for people to buy. This won't help if he refuses to attend events without her though, and it sounds as if that is a possibility.
3. A regular "friends' night" based around an activity the girlfriend does not like
This may or may not work out, but if you and your friend really like to do something that the girlfriend does not, you can try organizing get-togethers around that thing. The girlfriend may or may not tag along, but if she does sometimes she can observe for herself that:
- You and your friend are genuinely engaging in the activity (i.e., it's not just a cover for some nefarious purpose)
- She is free to come along but may not have a good time (because she's not into the activity herself).
As other people have noted, you can't make your friend engage with you in private. If he consistently chooses to accommodate his girlfriend's desire to monitor these aspects of his life you'll just have to choose between interacting with both or interacting with neither.