Down isn’t a great option in a damp cold environment in any case. In dry cold however, it’s unbeatable.
It's of course true that down loses some of its loft in humid conditions.
In practice, I found out that I never had any major issue because of that, though. I use a sleeping bag that is warm enough that even when it loses a bit of loft, it still keeps me perfectly warm. It shouldn't get soaked, but that usually doesn't happen (and a soaked synthetic bag is just as useless, though much easier to dry).
Down definitely needs a bit of care during a longer trip (air it out whenever there's dry / sunny weather, make sure it doesn't get soaked by condensation or rain...), but that's a normal routine after a while.
I think the "down sleeping bags are not good in wet weather" dogma is similar to "ponchos are useless when it's windy". There's some truth to it, but still, many pilgrims on the Camino prefer ponchos even in windy conditions, and according to surveys I read, most thru hikers on the AT use down sleeping bags/quilts despite often rainy conditions.
For ethical reasons, synthetic is of course the much better choice. Totally agree.
But regarding weight and sleep comfort, I definitely prefer my trusted down bag and wouldn't hesitate to bring it on a rainy trip, too.
A well cared for down bag lasts a lifetime, so, hopefully, I don't have to make that decision again!
To get back to the question asked:
Decathlon has not too expensive synthetic sleeping bags, but I wouldn't be sure if they're really warm enough in winter without testing them at home.
I used a snugpak sleeper extreme before, a bit over 2kg, rated for -7°C I think but in my experience only until 0/-2°C or so, but I'm a cold sleeper. Relatively cheap with 70€ nowadays (back then I payed ~60). Quality is okay. Maybe something like that could work without spending to much of the budget on new gear.