Hi and welcome to the forum.
I've done the Reiver's about 10 years ago on a cheap steel mtb, camping it and managed ok. But I did it in June and had a heavy thunderstorm just before Bellingham. Anyway enough of that, getting back to the real point.
TYRES!
What I would suggest is not slicks, but semi slicks. This enables you use both off/on road and there is some good tyres on the market. I think the Reiver's is mainly on tarmac and about a third is off-road, being roughtracks. But there is alternative on-road as a option.
Have a look on this link.
semislick