Is periodically transferring Bitcoin from an exchange to a private wallet cost effective?
But I know that moving it from the exchange to the wallet will incur a blockchain fee

There is a fee involved in all bitcoin transactions which goes to miners. However, withdrawal fees of most of the exchanges are not same as the fee used in bitcoin transaction for withdrawal. Most of these exchanges have fixed withdrawal fee like 0.005 BTC on Binance. Some don't have withdrawal fees like FTX (Maybe promotion and won't be free forever). Some exchanges allow you to set your own fee like Bitmex (Minimum fee changes according to mempool)
Spot exchanges also have their own fee structure for withdrawals. Peer to Peer exchange like HodlHodl uses 2 of 3 multisig so fees work differently and you don't keep bitcoin on exchange ever: https://hodlhodl.com/pages/faq

What's the must cost effective way of moving it to my wallet?

There are different types of fees involved in exchanges. You will have to explore all options and see what works best for you. Also prefer using an exchange that supports withdrawal to segwit addresses, LN and Liquid. Layer 2 solutions have lower fees compared to on-chain transactions most of the times.
I am going to make a list of exchanges that support latest bitcoin technologies soon. Until then this link can be helpful for exchanges that support LN: https://ln.guide/docs/EXCHANGES.html