[...] It does matter if the "dollar" value of BTC has doubled since then. Since if hypothetically the value of Bitcoin doubles after the halving, then there is still the same amount being paid for security
But since the value of BTC has doubled, the thing that you're securing has doubled in value, so you need twice the security. If a $5 lock is just-good-enough to protect a $100 bike, is it also good enough to protect a $200 bike?
Yes and because it has doubled in value, miners will be incentivized to increase their operations. So that $5 lock becomes a $10 lock in your analogy.
How would the miners be "incentivized" to increase their operations, after the block reward halves & price doubles?
Also, earlier postscript you might have missed:
P.S. To avoid complicated maths, model this at the limit, i.e. "the cost of mining 1 BTC is slightly less than market price of 1 BTC, as per satoshi's prognostication.
P.P.S: Which would you do:
1. spend 10 dollars to make five dollars
2. spend "five hundred and twenty million dollars" to make a billion dollars
(chose one)