You really need not worry about using the wood 45-70 in bad weather. Mine has seen rain, snow, and mud.. A fine coat of oil on the metal before you go out will help with rusting, then immediate cleaning once you get home will insure it even further. I coat my wood stocks with 3-in-1 oil during the off season. The wood absorbs a bit, helping to seal it off from moisture which causes wood to swell and sometimes change your zero. I wipe the excess oil off right before use so its not slippery.
A synthetic stock doesn't have much on a properly finished wood stock these days. Its more for personal taste and impact protection than bad weather. Wood dents, scratches, and mars easier than synthetic. The pallet wood on a Handi isn't a fine marbly English walnut that one would want to keep in pristine condition, but rather a utilitarian grade piece of tree just meant to be held and used. ...sorta like the difference between Russian Fabergé Egg and a Stanley clawhammer.
I'm not sure of the exact difference between the syn and wood Handi's.. I dont own a synthetic..