I get it, OP. I understand that no is sometimes not enough and you need to give a clear, watertight response. While I actually think parents can often do more harm than good by banning sleepovers with boyfriends/girlfriends, you don't have the room to do so in a way which is appropriate (shared room)/convenient (living room).
I think I'd go with:
"I've had a think about it again and..." (gives him a little bit of agency, at 16 I think it's good for him to see you taking him 'seriously', even if you don't have to)
"I'm saying no for several reasons..." (shuts off him trying to argue against one specific reason)
"1. You would either have to sleep in the bedroom with your brother. As it's illegal for sex/sexual behaviour to happen in the presence of a minor, I will not put you and Girlfriend in a position where you may be tempted to do anything of that nature in your shared bedroom. I will not leave the possibility of a crime occuring to trust" (it's not saying: I don't trust you, it's saying you're flat out not willing to put them in a position where you need to - that hopefully helps curb the argument that he can be trusted, as it's that you're not willing to need to take that on trust, as it's a legal issue)
"2. The only other option, therefore, would be for you to be sleeping in the lounge. I do not want to have to tiptoe around in my own home before work, nor do I want you having sex on my sofas and again, I am not willing to take that on trust." (Added the second part because his answer to the first will be - well, we don't care if you wake us up.)
"3. And lastly, Girlfriend is still legally a child, and I would not entertain having her stay here anyway without having spoken to her parents about their feelings and an honest conversation about contraception with all of us, you two included. That's to protect me as an adult from accusations of inappropriate childcare while having their minor child stay in my home, but it's also to protect her as a young woman. As it happens, there are other reasons that I would not be comfortable, but this would be a baseline requirement anyway and as this hasn't happened, it's a big tick in the no box" (might make him think twice, plus trying to hit the legal angle if he's firm on rules and rights. And if he's fully willing to introduce you all and for you all to have that conversation, great, you still have your other reasons and if they're genuinely okay with it then he can stay there!)
It's also worth reminding him that you are not infringing on his 'rights'. He has no right to be allowed to have anyone stay over at your house, and that whether or not he agrees, as a parent you have the responsibility and the right to make judgements that you think are correct as long as they're legal and safe.
...Having said all that, I think it's a hard one.
I actually think that occasional/not too frequent sleepovers with defined boundaries aren't a bad thing once teens are over 16 - say, no more than once a month, after an appropriate length of relationship after a conversation with everyone involved about contraception if necessary and sexual health as standard, with a chat about expectations for behaviour (come down for dinner/be up at a certain time/no inappropriate behaviour that could be witnessed or heard by anyone else/guest must be polite and engage with other household members).
It makes their relationship a part of family life which can/should be kept an eye on and discussed (rather than taking place entirely away from the home/on phones, which makes conversations about it harder for parents), normalises the sexual health/contraception chat, is part of growing their freedoms as they get closer to leaving home/adulthood and - and this is probably an unpopular thing to think as a parent, but it encourages better sex, and by that I mean it stops them shagging in half-built houses and alleyways like my friends did at 16, or quickly before parents got home. As girls, those sexual experiences (the hetero ones, anyway) were ALWAYS about the boy's enjoyment, as inexperienced teenagers with limited time/space are unsurprisingly not spending a huge amount of time unravelling the more complex beast of female pleasure, whereas - to put it delicately - a teenage boy is usually rather quicker off the mark. To be clear, I'm not saying, fix the orgasm gap and solve feminism by letting your teenagers shag at home, but my anecdotal experiences as a teenager says time and relative safety beats 'quick handjob under the duvet before his mum shouts you down for dinner and then you have to go home after' for giving you chance to not feel like something a teenage boy gets access to for his own enjoyment and yours is seen as too time consuming or complicated to bother with in the limited chance you get.