There are a lot of questions/comments in there, so I'll try to address them below. A quick tl;dr though: It will probably clear eventually, but the 5-gallon will probably take longer than the 1-gallons. If you can cold-crash it, it will speed things up. You can try clarifying agents, but they may mess up your plan to bottle carb. I don't know, though, I haven't tried it.
I was given the impression from my source material (The Compleat Meadmaker by Ken Schramm) and limited experience that it would clarify in about a month.
I ran into the same problem with the Compleat Meadmaker starter recipe. I ended up bottling way too soon, and those bottles now live in the fridge, so they won't harm anyone. For all the information in that book, it can be pretty vague in spots. Sometimes these things are clarified later on in little, one-sentence comments that are easy to miss.
However, even after three weeks, it's still pretty cloudy with no real discernible amount of change towards clearing at all.
Three weeks in secondary isn't all that long, for a standard, traditional mead. When I made that recipe, it took about 9-12 months before I declared it ready for consumption. It's still good mead, though.
I can still see some tiny bubbles at the surface. Is this evidence that fermentation is still slowly ongoing?
Not necessarily. Unless you actively degassed during primary, there is probably still a bit of CO2 in your mead. This will slowly dissipate over the next few weeks. If I remember correctly, though, that recipe has you rack early and it finishes out slowly over the next few weeks. The only real way to tell if it's still fermenting will be to check your gravity.
I'm just wondering, will it ever clear on it's own?
What kind of timeline am I looking at? Is this just the result of scale?
As long as it takes. Yes, larger batches take longer to clear naturally, in my experience. The amount of other particulate in your honey also matters. For example, I had a 3-gallon batch of mead from a super-waxy honey that refused to clear after about 3 or 4 months in secondary, even though my 1-gallons of the same recipe cleared in ~1. I bottled it, assuming it would just be waxy. 2 months later all the bottles were crystal clear, with a decent amount of sediment on the bottom. FYI, if you heated your must, it can take a little longer to clear.
[...] will I have to resort to clarifying agents? Or, because I want sparkling, should I bottle soon to ensure the yeast is still active and it should clarify in the bottle?
If you want it to be sparkling, then you should not use clarifying agents, unless you plan on kegging and force carbonating. Of course, you could always try to cold crash it, if you have the means to refrigerate a 5-gallon carboy. Also, keep in mind that if you've hit your yeast's ABV limit, then it may not carbonate when you add priming sugar or honey later on. Of course, there are solutions to that, if that's the case.
FYI, if you're concerned with the timetable on this one, don't get discouraged. While you wait for that to clear up, try some other beginner recipes to learn some more techniques, like staggered nutrient additions (SNA) and back sweetening. The BOMM (Bray's One Month Mead) and Joe's Quick Grape Mead are good, quick starter recipes that can help you build a base of techniques that you can carry over to other recipes.