Do you want both to bloom at the same time?

Trying to plant a clematis and a rose together and get them to bloom at the same time is going to be difficult to do unless you plant a rose that blooms over a long period of time or a clematis that does the same. When either of them will bloom will depend on the age of both plants--combos that bloom at the same time initially may not as either plant matures.

If you don't care whether they bloom simultaneously, then either type IIs or IIIs will work. Good choices for type IIs would be Konigskind and Piilu, both of which stay relatively small in size overall. Good type IIIs would be the heracleifolias, most of the integrifolias, stans, or any of the shorter shrubby types of plants like the US natives--fremonti, addisonii, ochroleuca, etc.

Just realize that whatever you plant, get the largest size pot you can purchase since the plants will need root run space or the plants will be dwarfed in size compared to what they would be if planted in the ground.

I was thinking of trying a miniature climber, which I know are sometimes only mini in flowers. So I plan to select one that only grows to about 6 feet or so. If I select a rose that has several flushes throughout the season, I was hoping that they would bloom together if I get lucky.

I don't know about a rose and a clematis in the same pot, just because someone gave me one--it was not successful and she was going to throw the whole thing out, but gave it to me instead. I took them both out of the pot, and the entire thing was just about all clematis rootball--it had completely filled the pot and the rose had almost nothing in the way of roots. Now a year later the clematis in the ground is growing well (not a surprise based on the huge rootball) but the rose is still struggling.

All four of my mini climbing roses are mini only in flower. Jeanne Lajoie is a monster. Wish I knew of some that were only six feet!