I haven’t grown sweet peas for a while, but I always used to start overwinter. They grow quite quickly so long as it is not too cold and the best bet is to keep pinching the tops out. That way you end up with short, stocky plants with half a dozen or so shoots. Once you get to spring, all those shoots will grow and produce flowers, so you end up with a lot of coverage.
As you are looking to produce a good sturdy root system, it is worth using a slightly bigger pot as well as feeding occasionally. I usually have onion seedlings and shallots and broad beans in pots early in the year, so have a bottle of liquid feed on hand. It lives in the cupboard under the sink so I can get to it easily rather than having to venture to the shed. Using that at half strength every couple of weeks really does make a difference to anything growing through Jan/Feb, even if you don’t think they are really moving much.