Dedication isn’t required for getting involved - many priests on-field will be very happy to talk to you about the Way, and you could do things like sit in on Assembly meetings and talk about the big religious stuff without either of those two things. Just saying you’re interested - not even on pilgrimage, just considering it, thinking about the Way and its place in your life - is, I expect, plentyWe like talking about religion, it’s why we’re playing priests. 
In particular, the Synod voted to make the Symposium open to anyone who wants to come; that’s a reflection of the general mood.
The main thing is you can’t vote on Synod motions; you have no official weight in Synod Judgements without a congregation. Still a perfectly good way to play and a string to have in the character’s bow, though.
However! In the interests of keeping options open, a refinement on the “non-priest congregations are strange” thing that might work for you!
If you did want that weight of “I can vote on this” without being a preacher, I can see ways around that sit in the sweet spot of unusual things that nevertheless reinforce the setting in really cool ways. The specific one springing to mind here is a Dawnish Questing Knight who’s the sworn protector of a small shrine somewhere; maybe one that has a relic of a Dawnish Paragon or Exemplar, or a relic of an unrecognised but super Virtuous person, or simply a shrine someone built.
The character need not spend all their time there, just drop in on it every so often. They don’t need to be a preacher - just the person associated with the shrine. If it’s a place a lot of the local yeofolk go to think about the Virtues, and if the character is the person they think of when someone asks “whose shrine is this?” then even without being a preacher or a formal priest, you’ve got a congregation. Just figured that might be on the right end of the priest/devout Knight divide to match what you want to play, while being close enough to the priest end to be a really cool way to flavour the requirements of being a Synod member.
It’d flow nicely into your Knightly Order idea, too; the leader of a religious community (even one who doesn’t preach actively) is another excellent less-priestly congregation style. Marcher Monasteries are the obvious place for that in the setting, but I’d take their existence as a good argument for similar ideas in other nations working.
(The difference in my mind between this and “pays for a church” is quite difficult to pin down; I think it may just be that “the shrine’s sworn Knight” feels more like a religious authority figure to the way I think, even without anyone specifically thinking of them as a priest.)
I kinda want to play this concept myself, but I’m a little booked up at the moment.
(I can see your point on the Congregation just talking about a church - I think it’s because, uniquely among personal resources, there’s some pretty major sociopolitical weight on who has a congregation. Other resources allow voting in some national elections, but the Congregation is the defining factor of one of the major Imperial political bodies, so it feels a bit more important to be solid on it in my mind.)
tl;dr: Congregation implies spiritual authority of some sort, but that could be more pastoral, monastic, or straigh-up informal than active preaching.
On Dawn and money: “can’t carry money” is pushing it a bit. It’s considered inglorious and demeaning for nobles to be concerned with money; flipping a Throne to a bartender so you don’t have to think about paying for drinks feels a little louche but not off-brief to me, and as a Questing Knight exploring the tension between the glorious ideal and the need to acquire things like food and boots when on the road alone could be really interesting.
you.