They won't necessarily allow a payment plan. It's not guaranteed, they've no obligation to provide credit, which is what a payment plan is. They can and sometimes do refuse to treat sick animals where the owners can't pay. People shouldn't rely on payment plans being available.
OP, if you've spent half your savings then you still have half left. So that's your budget for treatment, if there's no payment plan available. You tell the vet there's no insurance and have a conversation about what treatments options there are, upto and including PTS.
With insurance there's a clock ticking because they want things sorted out or not so they can exclude ongoing conditions upon renewal, so vets end up throwing all treatments and tests at the situation quite quickly. They also tend to go straight to the "gold standard" treatment because they know they can get the money from the insurance company and it won't affect you.
There's usually another more conservative path to take, involving a fair amount of trying this-and-that and "wait and see", because there's no time limit if you're paying. And the "gold standard" treatment might be quicker or better but often a conservative approach, whilst taking longer to resolve the issue, can achieve the same or a similar end result in the long term.
Ultimately thought, if the choice is expensive treatment you can't afford, the animal continuing to suffer without treatment or having to PTS, then you PTS to solve the welfare issue of the suffering. It's part of pet ownership that has to be faced.
Unless you're rich or willing and able to take on a bank loan or credit card to pay for treatments (personally I don't recommend this option, for your own welfare), then you won't always be able to afford to try everything before having a pet PTS. That's just the way life is. Although terribly sad, it's not wrong to PTS for financial reasons. It's better than the selfish alternative of keeping a suffering animal alive longer than it ought to be just because the owner isn't willing to let them go and face their sadness over it.
Your welfare matters too and it isn't sensible to run up tens of thousands of debt for any reason, but especially when the end result may be a pet that's no longer alive and you still have all the debt. That's why a conversation about treatments and their effectiveness and likelihood of success in your pet's particular case is important before agreeing to pay for anything. You have a lot more control over the situation than when using insurance. You don't need their approval to try experimental treatments, for example, because it's not them paying. So have an honest conversation with your vet and agree with them how to go forward.
You might want to look into whether psyllium husks (think that's the thing, you best double check) is any good for dogs. It's what's given to horses grazing sandy soil and at risk of blockages due to accumulated sand. Binds to the sand and helps clear it out of the digestive tract.
It's good news that the vets didn't think it an emergency and told you to wait for regular hours.