Maybe changing out the planter is in order - maybe get one longer so two plants can co-exist? Just a thought.

As for plants, I have two trellises on a south facing, but heavy west sun side. Last year I was 'fostering' a friend's four clematis plants and left them in their pots - they climbed the trellises (two plants on each) w/vigor and bloomed their stamens off. Cut them down after the hard freeze. Then my foster clematis became orphans (long, sad story), so I adopted them and planted them in the ground. They're very happy this spring .....

So, Clematis is up for consideration in your setting. (and, clematis will twine and climb w/just a bit of guidance)

Another idea for the exposure you describe is Carolina Jasmine or Jessamine (probably not the same, but marketed as such). I'm going to try this one on the east side of my house this year - I see them all over my 'hood, adorning mailboxes mostly, they seem quite hardy and they are evergreen.

Just a few random thoughts ..... am sure you'll get lots more suggestions and I look forward to reading them!

A clematis might work depending on exactly how much sun it does get. Does it get direct sun at any point of the day or is it all indirect sun? Clematis generally prefer 6 hours or so of sun per day although there are some that need less than others. If you do try a clematis, you might need a much larger pot for it since I can't gauge how large the pot is. Generally for most robust vining clematis, minimum pot size would be 18 inches tall by 18 inches wide. I personally like to go bigger for the ones I have growing in pots.