I have used Flutter but not with DAML which is what I guess you are after.
I actually have a similar question and that is regarding the use of JavaFX for the UI layer.
DAML recommends React as part of its UI stack component but the little that I know is that apparently the gateway is thru a JSON/RPC gateway so perhaps one doesn’t need direct support. Flutter does have a JS integration package i.e. which should allow one to tap into the DAML React libs.
I do like Flutter but for this scenario I would probably in my case prefer to use JavaFX and thus my question is whether the DAML Java bindings are sufficiently complete as the React bindings. For example, is there a full blown ledger API available from the Java bindings that I could therefore tap into from a JavaFX/Groovy UI ?