There's an oxymoron in your question, namely 'climbers' and 'small containers'. Flowering climbers will need large containers, or they won't be able to reach any height at all, I'm afraid.

Well , there are several very enthusiastic annuals that you could try , morning glory ( I had a red version as well as the blue last year and there was no stopping it !)
The wonderful nastursium , and many others I'm sure , but as far as perennials in your area I couldn't say , but good luck , anyway .

Passiflora do pretty well in smallish pots. However it can leave the roots susceptible to hard frosts.

Aserina scandens is an annual that grows well in pots too.

Gloriosa is ideally suited to container growing. Dry the pot out in early autumn and shove it in the cupboard under the stairs for the winter.

I have a jasmine clotted cream in a pot, there is another jasmine which is evergreen called star jasmine which my mother in law has in a pot and that looks good, I am not too sure how it would fare in a small pot. Sweet Peas are fab but they are annuals.

How about one of the nightshade variety? Very very hardy, pretty flowers and lovely (poisonous) red berries all autumn and winter. I cut it back to ground level early each spring. It can be invasive, but only if you don't cut back to base those sprouts you don't want - every couple of weeks when you are in the garden check growth and remove as appropriate. It needs a trellis of some sort or netting to climb up by entwining with it. Don't know the name of the variety I have but there are several to choose from - check climbing nightshade on google. If you have problems then I may be able to send you a root. Let me know.

Thanks to everyone who responded to my question re container climbers plenty of ideas and I will give some of them a try.
Thanks again Costus