Bank account balances aren't money, they're IOU's, a record of debt - a bank handing out a statement that shows an account balance of $100 quite literally means the bank saying "we acknowledge that [as of date X] we owe you $100" and nothing else.
The standard money transfer from Bob to Joe is a deal where the bank says "ok, Bob, we owed you $100 but if you want that then we'll now owe that $100 to Joe instead".
It's also worth noting that's just a record of debt not reality - there has to be some legal basis for that transaction to actually change the liability between the bank and the account holder, simply changing the balance in a database doesn't change the amount of debt but just the record i.e. "bank's opinion" of that debt; and if that record/opinion is wrong, then that balance can and will be disputed, and if the dispute can't be resolved otherwise, then it'll be up to courts to decide if that debt is valid or not.
If you record just the credit without the debit, then it's the equivalent of the bank unilaterally agreeing to new debt, the bank asserting that it now owes $100 to Joe just because. It's free to do that, but it would mean that it's "books won't balance" i.e. their accounting isn't consistent with itself and doesn't match reality, so to properly account for that transaction they'd have to book a debit to their profit&loss statement since they lost money by acknowleding that balance increase i.e debt without an offseting balance/debt decrease to someone else.
The sending bank will have their Bank of England settlement account debited by the central FPS system, based on the central FPS system's view of the world, at the end of the settlement cycle.
The recipient bank will receive the money into their settlement account at that time. If the sending bank doesn't debit their customer then both sending and receiving customers will have the money in their accounts, but the sending bank will be out of pocket.
The enforcing entity here is the central bank they both have accounts with. If a customer of Monzo sends £10 to a customer of RBS, then that money never leaves the central bank, and both records are just updated. But the total amount in the central bank must still add up to the accounts of both Monzo and RBS, otherwise there is a discrepancy.
The settlement process through a central bank is a way of ensuring that banks dont need to literally send truckloads of cash to each other at the end of the day.
Monzo says to the central bank, "today I sent RBS £1,500,000", and RBS says to the central bank, "today I sent Monzo £1,200,000". So the central bank just debits Monzo's account with them by £300,000, and credits RBS's account with them by £300,000. The total amount in the central bank remains the same.
So, sure, a bank could claim they sent less money to another bank than they did, but eventually the numbers wouldn't add up, and it would trigger a bucketload of auditing, likely resulting in revocation of banking licenses, and legal issues for both the bank and people involved.
There is also a technique involving things called nostro/vostro accounts, where banks have money on deposit with each other, and the sending bank's deposit with the receiving bank is used to cover transfers:
Of course, then they need to keep their accounts topped up, and they can do that by transfers through other banks, which might be central banks or commercial ones. The nostro/vostro system is suitable for use where banks don't trust each other so much, eg because they are in different countries. I think it was used more in the past, before reliable central settlement schemes were established, but i'm not sure.
You can think of net settlement as being a bit like nostro/vostro where the accounts have infinite free overdraft facilities, and so the banks never build up a credit balance, and just settle their debts at the end of the day.
