inner-exchange "fee's" are % based not fixed amount.
Depends entirely on where you are buying. Last I checked, Coinbase for example (not Coinbase Pro) charges a flat fee of $2.99 for a trade of $200, but a 0.5% fee for trades above $200. So 7x $200 trades would cost $20.93 in fees, whereas 1x $1400 trade would only cost $7.00 in fees. Gemini have a similar structure with flat fees for small trades and percentage fees for larger trade, which again works out better if you make a single larger trade rather than multiple small ones.
Still, this is not a perfect way to save the fee while the dollar cost average. For example, let us suppose bitcoin prices start to pump for the next 7 days, so if OP buys daily he will get more satoshi's as compared to if he bought all in on the seventh day when prices are high. In this case, the saving fee may not be that important.