I somehow don't see Gallia being able to beat the Dragons in their Chugoku system. They are masters of hit and run in the shifting clouds and amongst the constantly moving asteroids there. Gallian tactics would have to change radically to be able to operate in that environment and large scale operation are not feasable there, it would have to be small strike team work using only fighters as even gunboats are at a distinct disadvantage there and capital ships would be virtually helpless in there and anything other than small teams would be to unwieldy to control in such an environment. Sending in small strike teams of fighters against the Blood Dragons in their own territory where they train every day, is something the Dragons would look forward to with great pleasure.
As regards finance, the Blood Dragons have means of finance in place that are not common knowledge, even amongst the Dragons themselves. These include finance from Dragon sympathisers operating inside Gallia already as well as certain individuals running commercial businesses of all kinds, both lawfull and unlawfull, in various parts of Sirius who regularly contribute to the Blood Dragon Coffers. So continuing finance would not really be that much of a problem for them.