As a nonnative I'm not 100% sure I'm nailing the tiniest nuances correctly here, but here goes anyway.
The word ちょっと I think is taught as a way for lower level Japanese students to deal with a wide variety of situations without having that broad of a linguistic repertoire to handle specifics. That said, if you respond to someone's invitation just by saying ちょっと, the point will get across, although it is not necessarily the most polite way in this situation.
To get to the meat of your question first, basically if you want to refuse something extra politely you can just state your reason, but the intonation is pretty important. Like, あ、すみません、土曜の夜はもう予定が入っていて・・・ you can leave off the 行けないです or ちょっと無理です or whatever and just end it on the て form, but you need to sound sorry about it!
If it's someone you're familiar with, it should be no problem just casually saying that you can't do something and give a reason.
Now, to make a note on ちょっと: I don't think it's wrong per se, but it's not what I would call polite either. More neutral or even distant depending on the circumstances, like "I can't/don't want to and that's all I really care to say about the subject." For that reason it may be hard to ask someone why if their response is just ちょっと.
I could be wrong about the negativity part, but I do associate ちょっと by itself with some more negative usages. For example, if someone does something dumb or unpleasant, you could say ちょっと！ to mean like "hey!" or "cut it out!" Or you can also use ちょっと to say that something is a little off or unusual in a negative way, like 「昨日上司が私のお尻を褒めたよ。」「それはちょっと・・・だね」 I think it's pretty close in English to saying "that's a little.. yeah," to use the previous example.
You don't need to avoid ちょっと, but it's best used in conjunction with other things. To my ears saying something like 「それはちょっと無理かもしれないです」 is pleasantly vague with a tinge of regret (depending on tone of course but yeah) whereas 「それはちょっと・・・」 is a little colder and potentially condescending, as if to suggest it's strange.
At the end of the day you're a foreigner and people will trip over themselves to accommodate your poor Japanese (even if you're speaking fluently many people still treat you as though you're speaking in 片言 and simple/wrong speech...). The ultimate piece of advice I can give on the subject is that foreigners don't need to concern themselves with the nuances here since I'm sure you would never be misinterpreted as being rude when it wasn't your intention.
Anyway this got a little ramble-ful. If I'm wrong about anything please let me know and I'll edit or scrap the answer entirely.