Both Stoßgetweete and Stoßtweet have their justifications. I stumbled across the form Stoßgetweet but I can totally see its (admittedly much weaker) justification, too.
Let’s start off with (the singular word) das Stoßgetweete. The site is called Twitter, the German verb for to tweet is tweeten and in the same way as transforming reden into Gerede or fragen into Gefrage, we can derive das Getweete for a series of random, unimportant, unnerving, or whatever else you want to add here tweets. If that happens ‘shotwise’ (stoßweise, e.g. #Aufschrei), you can be inclined to call it Stoßgetweete. I wouldn’t use it in a positively worded sentence.
Now a single tweet, would be just that in German, too: ein Tweet. Don’t ask me whether the verb lost a shwa or whether tweeten was derived from the noun. All that I can definitely note is that contrary to nouns like Frage, Rede, Bitte, etc., this word does not end with an e. If you blurted out a single tweet in a similar way to the wave of tweets mentioned above, that could be called a Stoßtweet. (Or maybe it was just a Tweet that was supposed to shove somebody/something in a certain direction, up to you to decide!)
Let’s finally consider Stoßgetweet. I do not consider this a well-formed word, unlike Stoßtweet and Stoßgetweete, which I would immediately accept. However, it kind of makes sense in the context you give by alluding to the Stoßgebet. I would consider this a malformed word for generating effect, though. I doubt too many people would have connected Stoßtweet to Stoßgebet while Stoßgetweet immediately suggests such a connection.
The thing I dislike most about it is the seemingly extraneous -ge-. As mentioned above, I deem Stoßtweet to be a good enough word and I don’t deem Getweet to be acceptable. The prefix ge- is a pretty common one in German, especially due to it marking past participles. I’m not sure about whether the ge- in Gebet is one of those, though and that is the weak basis that that analogy is built upon.
tl;dr: No, you cannot choose your grammatical rules to make up words, but if there is a sufficiently strong analogy, you can get away with some misformings better than with others.