Doubling price is not at all realistic solution because rewards are ultimately going to become zero in next century.
By logic of doubling prices , In next 40 years ,BTC should cost more than 7 million USD per BTC. I just wish if it can be true.
The block subsidy becomes less relevant with each halving, as it slowly phases out in favour of mining fees. About 4 halvings from now (ie. in about 16 years) we're looking at a block subsidy of 0.78125 BTC which is less than the transaction fees per block during peak times. 2-3 more halvings after that and we're at a block subsidy that roughly equals current transaction fees during calm times.
Regardless of that Bitcoin's price doesn't matter all that much for network security as long as Bitcoin remains the largest coin by the fiat equivalent of block subsidy + transaction fee within its PoW scheme family (ie. as long as Bitcoin is able to sustain more SHA256 miners than alts that share its PoW).