A doubling leads to the rewards staying about the same. Whether that needs to happen for bitcoin to survive is not at all obvious. It would likely be sufficient for bitcoin to survive, it just isn't clear that it is necessary, maybe ⅓ of the reward would keep folks interested.

It probably needs more than doubling in the long run. For the following reasons.

Price is measured in USD and USD is inflationary so its purchasing power drops and BTC needs double the purchasing power not double the USD "value number"

The reward being constant would in the long run reduce the margins used as profit but also stop the re-investing of said profit to creating new hardware (ASICs) etc. That would stop or slow down the growth of hash/energy ratio which would make 51% attacks cheaper over time if we assume all other hardware on this planet keeps getting better and more.

Also keep in mind that BTC "only" doubling ever 4 year would significantly under perform most peoples expectation so if it's even lower, interest probably goes down rapidly as well which could trigger the collapse.