Another recruit to the darkside!
I’m not putting old cars in my front yard though! I’ve done that before and it didn’t go well. 
Could you just get some grass seed that’ll grow in the winter and spread that around? That might be the easiest and most cost effective route to take.
Then if the dogs wear paths through it, then you can always carpet those areas, etc…
Reminds me of my mom talking about the house I grew up in. They moved in when I was 13 mos. old. This was in December. This was a brand new housing development, so there was no landscaping yet. And it was a rainy December. AND they put the clothesline in the southwest corner of the yard, when the only egress from the house to the yard was on the northeast corner of the house. She would have to schlep the laundry across a sea of mud to the clothesline (until my dad built a better one much closer). One time she said she turned around to see me crawling through the mud to get to her. I think that’s when dad built the new clothesline.
This would have been my suggestion.
An annual rye grass will come up in less than a week, is fairly cheap, and can be replaced fairly easily when you get some funds.
If you weren’t living in explosive fire land I’d suggest you locate your local county chip drop program and just put out about four to six inches of chips, preferably cedar (because it repels fleas, bonus!) in the problem areas. Costs nothing but a bit of time with a wheelbarrow and rake, keeps the dogs’ feets from getting too gross and they’ll mash it all down into the mud by spring when you can decide on a more permanent solution. It’s good mulch and turns into good compost too. The only downcheck is that in summer when it all dries out the remaining chips will become tinder dry, which is not optimum near the house.
I was thinking about some kind of mulch, but the husband pushed back.
It does have the advantages of being both cheap and easy–maybe just the worst trouble spots get chip mulch to tide you over to spring?
This sounds dirty to me.
Our dogs tore up their portion of the backyard (which is fenced for them) in early spring. The ground would freeze at night and then thaw during the day but it would still be damp. There was no grass left after a few days, just mud. I was going crazy with 12 muddy paws every time they came into the house. We threw down wood chips. I’d much rather have a nice grassy area, but the wood chips do the job. It’s fairly cheap and we just throw on another few bags when it starts to grind down into nothing. It’s probably not a good surface if you’re using a cane or are not sure-footed.
Well. . . we just put pavers where we needed to walk, and otherwise just stayed out of the yard.
That’s was I would do.
We also use wood chips. Our house is so much cleaner since we did this for the wet months.
Wood chips are in the lead.
Of course, now that I am planning for this there is no rain in the forecast here for the foreseeable future. Drought looks like it’s winning. 
But I hate that we no longer have grass. But if I have to choose between grass with muddy paws and wood chips with no muddy paws, I’ll choose the wood chips!