What should I use to produce the hotness of Hong Kong-style borscht?

Of the recipes you link to only one contains paprika, another one lists "Chinese chili paste, chili oil, hot sauce or dried red chili flakes to taste". The one that specifies paprika doesn't mention a specific type of paprika. Without further distinction, that generally means the sweet, non-smoked variety. Sweet paprika is just dried, powdered pimento, the type of pepper seen most often (here in the US anyway) stuffing cocktail olives. It's not hot at all. Hot paprika usually gets its heat by mixing in another, hotter, pepper. Even those paprikas are not generally overwhelmingly hot.

Paprika of any type (hot, sweet or smoked) from any country (Hungary is known for it, as is Spain) is going to bring you closer to a flavor profile more reminiscent of borscht's Eastern European roots. You may want that, but I wouldn't count on paprika to to provide the kind of heat you're talking about.

For Chinese hot and sour soup the best sources of heat are white pepper and chili oil. Nothing else tastes "right" in my opinion. With that in mind I lean toward those ingredients for your borscht as well. You can buy chili oil in any big grocery store, but Asian markets will have a greater variety at probably a much lower price. You can also make your own.

So I'd recommend starting there. White pepper, chili oil and paprika. Make a batch of soup with none of the above and add each spicy ingredient to samples of soup until you hit the combo you like. For best results, once you got the ratio of spicy ingredients figured out, scoop out a bit of your broth and simmer the spices for a good 30 minutes or so (the pepper and paprika especially, it doesn't matter so much with the oil), and add the simmered spices into your cooked soup. They're just better cooked a while. Next time you make the soup you can add the spices in the beginning, since at that point you'll know what you like.

Oil is the optimal medium to bloom the taste of a pepper. For that reason I would go with the chili oil or paste. It will be easier to control the amount of heat in the soup because it will disperse more quickly and you can taste test more easily.

If you choose to use paprika, I suggest getting one that is labeled "hot", but not one that is labeled "smoked" Smoked paprika is very good but has a distinctive flavor profile that doesn't look like it would fit with a Hong Kong dish (it is super in Spanish food though)

I have had very good luck using Roland Curry Red Paste. I assume it is available in ON. I am in Chicago; it is available in the mainstream supermarkets and certainly the international grocery stores. It doesn't taste like Indian curry as Midwestern American's know curry, but is a pepper paste with a number of other spices in it which gives a nice complexity. I use it in a chicken broth based soup similar to the recipes you posted but without the cabbage. I also use it to mix with mayonnaise for chicken salad.

My jar says it is a product of Malaysia so most of the ingredients would be available in Hong Kong and therefore consistent with the flavor profile you are trying to emulate