I agree with your grammatical point but I think there is an evolution in the use of english going on here that gives Goodgirl some room for expressing herself in the way she is inclined to. Certainly in England it is quite commonplace nowadays to describe sport rather like an event or experience and thus use sentences with the article, like :
He's at the tennis. We went to the football. Thanks for coming to the cricket with me. I enjoyed the bowling, didn't you?
Yes, if you were watching a bowling match (e.g. not sixpin 'American' bowling) it would be "thank you for inviting me to the bowling," but if you were bowling yourself "thank you for inviting me bowling" would be the most natural, especially within the informal context. ( "thank you for inviting me to go* bowling" would be more correct, but it sounds less natural here)
*Bowling is a sport in English which takes tends the verb 'to go' (or 'to do') (e.g. to go bowling, to go swimming, to go running, to go rock climbing etc.) instead of the verb 'to play' (to play rugby, to play tennis etc.). So it would be "thank you for inviting me to play rugby" and "thank you for inviting me to go bowling/swimming/running" HOWEVER because these end -ing, in an informal context the 'to go' can be omitted as I showed.
Thank you for inviting me to go bowling = thank you for inviting me bowling (informal)
Thank you for inviting me to go swimming = thank you for inviting me swimming (informal)
Either Thanks everyone for inviting me bowling, I had a great time or Thanks everyone for inviting me to go bowling, I had a great time would be perfect.
It sounds like the context is informal anyway, so I wouldn't worry about tiny grammar details - sorry my post was rather long