For Pressure Canning, can I use any other jar than Ball’s Mason Jar

In answer to your second part, no. If something is not acidic enough for water canning it seems it doesn't matter how long you process it, it isn't safe. This is the reasoning all the cookbooks I've read provide for adding a teaspoon of lemon juice to each pint of tomatoes.

I would therefore use this strategy:

• get things to the right pH for water or pressure canning as a separate first step. (Fruit will generally be ok, tomatoes need lemon juice, pickles will be ok, etc)
• check the processing time for pints and quarts of various things (jams, fruit, pickles etc) and look for the pattern: is it consistently 25% more time, or 5 extra minutes? Work out the pattern.
• repeat for hot vs cold pack: do you halve the time, or subtract 10 minutes, or what?

Now armed with a hot-pack pints recipe, you can adjust it for cold-pack quarts or vice versa. You should also, in poring through these recipes and charts, have come to understand various categories (fruit, jam, pickles, nonpickled vegetables etc) and be familiar with what times they need. Then faced with a chutney, conserve, or pickle that isn't on anybody's list, you should be able to choose what category it most likely belongs to, and safely pick a time for it.

I'll leave the question of food safety and mandatory canning or not with regard to ingredients to others, but frankly, if you have a large pot in your kitchen you have a canner. For small batches and only very occasional use, there is no need to buy a separate canner.

Place a small metal rack (if you have one small enough) or a folded towel on the bottom of the pot (I like to use my pasta pot with the insert), put jars on top and fill with water until almost to the top of the tallest jar. Yes, you can process jars of different sizes in one batch. Rule of thumb: cold jars go in cold water, hot jars in hot water. This setup is equivalent to a separate canner minus the extra gadgets like timer, thermometer / thermostat and other features that are for your comfort and ease of use, not essential for the physical process of heating and sealing the jars.

So if you want to play it safe, put your sauce in the jars¹, add lids and screw them on with the bands "fingertip tight". Place them in the water bath and slowly bring to a boil. Once your water is simmering, "canning time" starts. Off the cuff, ten to fifteen minutes should be fine for a cooked BBQ sauce, longer isn't a problem either. After that, lift the jars out and cool slowly in a drought-free area. If done correctly the lid will stick to the jar even without the bands, so you get an extra safety mechanism: If your sauce should spoil, the lid will come off, being a clear indication of trouble.

But my main reason why I suggest canning the jars is this:
If you ship your sauce, you have no control over the storage conditions of your jars during transit and - depending on the destination - constant refrigeration or similar might not be possible everywhere on this world. That alone would be reason enough to do the extra step, which is surprisingly easy once you give it a try. I suggest leaving the rings on during shipping and instructing the recipient to remove them on arrival to notice any signs of spoilage.

¹ This includes the usual steps done even without canning, e.g. cleaning the jars and lids, "sterilizing" them in your oven or in boiling water, keeping the rims of the jars clean and filling the hot food immediately into the jars. In short, what you would normally do to ensure general cleanliness and food safety.