Not got a cx bike, but regularly ride the fire roads on my 29er with 700C’s.
As stated, in order to avoid endless “slashes” I have switched to tougher “touring” type tyres, e.g. conti travel contact, schwalbe land cruisers.
Also as stated above, anything off-piste will be vitually impassable at the moment. Likewise bridleways around Downton, Alderbury etc. The river has been really high for weeks and all surrounding fields/paths etc under water.
I think you’re best bet is a fireroad/road combo. Sounds dull but the scenery is nice if you get a good day. It will be hard going in places as even the gravel tracks soften up in the winter. But not impassable/impossible.