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.