Slicks are usually only allowed as long as you have rollover protection. In the case of a coupe, it's built into the car so you'll be fine (the rule is more there for Caterhams without a cage).
Personally I've never run slicks myself so can't comment too much, however depending on your track day experience I would give very serious thought before using them. Yes, you will get a hellish amount of grip with them, but as Amstrange1 says above you need to have a good understanding of how they feel and how important warming them up and cooling them down are. They will also allow you to put far more lateral load into your suspension parts as you correctly assume, and whilst I don't feel there's any particular weakpoints in the Zed's suspension that shouldn't be able to cope quite happily with that, it's just worth bearing in mind.
A road car on 888s will be more than quick enough to be fun at Oulton, and even doing it on ditchfinders will be fine and arguably more fun (I've got Event tyres on my MX-5, and it was hysterical there in October). I wouldn't bother putting slicks on unless it's something you really want to try.
Also, it's T-Y-R-E-S.