Hey fellow gardenstater! I live in OH now but grew up in NJ near Asbury Park! lol
To help with you question...
I am trying to do the same thing here with a new clem Henryi, and several others and my John Cabot rose which is coming in a couple of weeks.
I am also constrained by space because they are against a trellis.
Since 'Haku Ookan' is a group 2 clem you might want to consider the two side by side? Think in terms of not only water and nutrient requirements, but which way will be easier to prune both? Clems don't really need to have "their roots shaded" as what they really need is Moisture! The roots shaded by mulch or low growing plants (or the rose for that matter) is only there to keep the clem and rose moist. If you can't plant them side by side I would tend to think in terms of planting the clem in front of the rose.
Of course, it is all up to you. Our gardens should be done the way WE like them :)
I think at least a ft is ok for the spacing if you are training it to go through the rose. But be careful digging as all roots are easily sliced through.
Mine are also going to be about a ft away from eachother.
Here is a pic of the general plan:
(there is actually a lot more space than what is shown as the trellis is 20 ft long and 6 ft high.) But I'm not that great at drawing lol.

{{gwi:583181}}

So do you miss ol' Jersey? It's funny, I'm just a few minutes from Asbury Park in Shark River Hills. Grew up in northern NJ, but wound up down here when DH was transferred. It's flatter than I like, but I did find a little hill to live on, just a few blocks from a tidal marsh, so it is beautiful and a great spot for birding, and I do love the growing seasons here.

Thank you for your advice on the placement. I may just put the clematis in front and slightly to the left of the rose (the side away from the arbor), just so I don't tear my hands and arms up while trying to plant (and subsequently prune) it. I'll probably dig the planting hole with trowel and hands to avoid, as much as possible, damaging the roots of the rose. It would have been better to plant them simultanously, as you are doing.

That is going to be some awesome display you have when it's all done! I hope we'll see pictures.

And I agree, our gardens should be the way we like them ... if others enjoy them, too, that's icing on the cake!

I plant mine about 4-6 inches away from my climbing roses..Jeanne

I grew up in Little Silver, near Red Bank. Came out here with DH. I like it here though. Lots of rolling hills! Some farm land, but mostly residential homes and rolling hills. Not half as much traffic either. But, I DO miss the beach like crazy! I used to love walking the boardwalk, especially for exercise. And you have much more and better restaurants around. Most restaurants here are around Cleveland, which is about an hour away, and can stay an hour away lol.
As I said previously, you have to do with the space you have and what YOU like, and at the same time thinking ahead to what will be the easiest for you to work with in terms of pruning, feeding, pinching, watering, etc.