I have a single seat gyrocopter with rotax 912uls on it.
I experienced same problem on one carburator where after shut down fuel was flowing out of vent tube.
I took both sets of floats out of carbs without removing carbs from engine. One set was way over 7 grams. This was the one set from carburator where fuel was coming out of vent after shut down. I am sure probably also on occasion fuel probably was coming out during flight especially maybe during low power during landing. I use 100LL avgas, aviation fuel all the time.
Yes floats are expensive. I believe 150 dollars a pair. I replaced both sets since the other set though within weight limit was getting close to probably getting heavy.
Suggest as others have advised start with the floats and see if this will resolve your problem as it did mine. They claim the new version floats are better but rotax does not make carburetors. Carburetors are manufactured by Bing for them and parts. This float problem has been an issue and I don't believe a scientific explanation has been provided yet.
The person I bought my gyrocopter used none alcohol car gas. Car gas is very unstable and when aircraft sits for a while especially during winter this may be the reason floats get heavy. I may be wrong on this. Suggest running aircraft on aviation 100 low lead for the last flight if one puts aircraft away during winter or other reasons.
Because of car gas people usually run there lownmoyer empty before putting it away for the winter. This is not the case regarding aviation gas 100LL. You lawnmower will start even if aviation fuel is left in it for several years without touching it.
Don't use aviation fuel all the time in 912 unless you are willing to follow suggested procedures since engine oil is shared by the gearbox. This means changing oil more often, cleaning oil tank, using chemical in gas to scavenge lead like a product called TCP or similar, using oil that is not all synthetic, and changing plugs more often.
I personally change my oil every 15 hours and clean the tank. I know it's overdoing it but oil is cheap compared to engine and gearbox maintenance. I use aeroshell recommended oil but probably am considering using pure, no synthetic oil in it, like motorcycle Pennzoil 10w40 since I only use 100LL aviation fuel. Synthetic oil and lead In gas don't get along. In my state none alcohol fuel is very hard and far to buy.
As stated in maintenance issues, start with the simplest and see if it resolves the problem. Hope it's the floats though parts expensive labor not so much.