Would you recommend staying in Taxim in order to get to the track easier?
Yeah absolutely. Rings will be departed from Taksim Square at each 30 minutes or so. It will be way easier for you to stay in Taksim.
Furthermore, to me, Taksim is way better than Sultanahment also in terms of accomodation but of course it is all down to you. Sultanahment is not far from Taksim, so you can go there if you want even on foot on such a shiny beautiful İstanbul day. It's really nice to walk on the way between Taksim and Sultanahmet (On the way there, You'll see Tunnel, Galata Tower, Golden Horn, Spice Bazaar etc.). It'll take your 30 minutes to get there. And you know what, in Sultanahmet there are more tourists than Turkish people
So you won't lose anything to stay in Taksim. It's much cheerful, energetic and exciting. (Tbh, I get a little bored in Sultanahmet but of course you should see there too).