It's silly to say "so he doesn't want to live with you but will with a stranger" 
A part time, lucrative (£400 a month tax free!) stranger can be asked to leave anytime.
Vs a girlfriend who'll be there full time and who it will be awkward as fuck to deal with if he changes his mind about being there.
Yeah, that's the same 
It doesn't matter that she's female.
It doesn't matter that he wants to give her the first night without you there (I think that's good manners)
It doesn't even matter that after 2 years he doesn't want to live with you - it's not that far into a relationship.
What does matter is that you're all wishing and hoping about living together and too scared you'll lose him if you do anything other than mope about day dreaming about it.
If you really think your relationship has legs, say "this lodger business has really made me realise I hoped we were moving towards living together in the very near future. That's what I want. When will the lodger move out*, and shall we live together then?"
*maybe it's a short term posting at work, otherwise it would be reasonable on her to make it only 6 months, or discuss with her that you're moving in so she needs to decide if she can cope with the dynamic of couple + lodger