Thorny budgeting question

This is sort of related to a cash-only basis existence, but it’s got bigger implications than just that. So here goes. We’re really scraping every month to achieve a zero-based budget. Some months we do, some months we don’t. I am a HUGE fan of sinking funds, and have ID’ed quite a few categories where we KNOW we spend money on an infrequent or irregular basis. So far we haven’t set up sinking funds for those, but I’d really like to. Yet by doing so, and by diverting even a smidgen of money into them (we’re talking less than $10 per category per week), it’ll be even harder to hit our zero-based budget every month.

My DH is very reluctant to invest in sinking funds for that reason, and prefers to put off the cost and taking it out of savings when the time comes. Of course when the time comes, it stinks to take it out of savings when we could have had it ready to go. Sometimes, a bill will go unpaid for awhile because we hate taking it out of our general savings. That tells me we really need a sinking fund for that item, but he’s resisting because it’ll blow the budget.

This has actually turned into something of a sore point between us, and I’m wondering how others have handled this. It feels to me like he’s just afraid to commit more money to more things, but then hates to spend the money when the time comes. It’s a no-win scenario, which I’m trying to change, and he’s resisting because any alternative still costs money. I just don’t see a downside to making the switch but I’m getting a lot of pushback. Suggestions?