Sorry, this is going to say what you don't want to hear.
Can anyone give me any advice on whether I should be submitting my support for his EOI anyway, or whether there is any chance that we could apply for a Partner Visa even though we haven't lived together?
No, there's nothing you can do to help his EOI, and no, you can't be a partner in INZ terms without having lived together.
The mention of 'partner' on the EOI, giving extra points for the partner's qualification, is for when a couple are both applying to immigrate, one as the main applicant and one as the secondary applicant.
For the purposes of sponsoring someone in, a partner is ONLY a partner when they can prove the relationship in the terms laid down by INZ. There are old threads here
https://www.google.co.uk/#q=site:+en...tnership+proof discussing what is required. You need 2 -3 months' evidence of living together for the application to be taken seriously. Yes, you may well have to arrange to be together with one of you on a visit visa, or go travelling in places where you can both legally be (since you don't have to have a fixed base to prove partnership - if you can show that you were both together as a couple using shared accommodation, that will work, too).