The bank surely won't let him keep the house. Even if, by some miracle, Chris did get to keep the house, he likely wouldn't be capable of taking care of it anyways.
It's probably best if Chris sells the house, gets whatever equity out of it that he can, and gets an apartment or splits one with some roommates.
His tugboat might go up a bit, but who is going to deal with the responsibilities of adult life with Barb gone? He always has been very oppositional towards help, so its not like any money managing advice would get through to him.
Chris isn't that bad at managing money.
This is how it works: every month the tugboat comes in. Barb takes her cut and Chris is left with the rest to do with what he will, which includes paying his bills. Chris has been on this regimen since Bob died. It's not like Chris sits down and writes out a budget on paper, but he does do some basic estimates in his head about his financial state. So at any given point in the month, Chris does maintain something of a running total in his head of his situation; how much money he's got, how many days are left until the next tugboat, what bills he's got to pay, etc.
Chris is a dumbass, but even Chris knows he can't bitch the electric company into giving him free electricity. If Chris gets a bill for something on a regular basis, he's capable of remembering it and paying it. He's been doing that since Bob died.
This isn't a responsible way of handling money of course. It's like redneck financials, where you don't do any planning and you only pay the minimums, and you only think about them when they physically show up in your mailbox. But it's not the level of incompetence that reading the cwcki would lead you to believe. Remember: the cwcki documents facts, but it's not an unbiased perspective. It's a humor website, not a documentary.
Now what's going to happen after Barb dies, is that he's suddenly going to lose her half of the income (she's got a pension and social security or something). There's a big range of possibilities for how well he survives without her, depending on if he's got someone to steer him into a sustainable lifestyle or not.
I think trying to keep the house would be a mistake because even if he can sustain it (which is debatable), he'd be constantly stressed for money.
In fact, whichever route he takes, he's going to be constantly stressed for money. That's one thing that's going to happen regardless. Right now, Chris is actually in a a pretty comfortable situation, financially. If you don't give a shit about being a lifelong debtor (and rednecks never do), then Chris just lives a lazy life with barrels of disposable cash.
Yeah... when Barb dies and he's footing all the bills alone, he's going to feel every little bump in his finances. The thought of getting sued or being a little late on a payment is going to scare the shit out of him. And that will be the rest of his life. No more toys. No more fun. Just being all alone, being completely unprepared for life on his own.

, but there is always the hope that Barb dying would be healthy for Chris in a weird, twisted way. Once he sees the cold, uncaring grip of reality close in he might realize that there is nowhere to turn other than being personally responsible for a change. It might finally force him to become somewhat of an adult, as he will no longer have anyone to enable his childish ways. Maybe Chris could manage living in an apartment with his tugboat money, possibly with a few roomates. He'd have to learn to be a tolerable person to be around, along with everything else, but when the options are grow up or starve he might change.
I think he'll survive, but I'm not super confident of it though.