Bournemouth itself doesn't seem to have many seafood restaurants. Even the ubiquitous Loch Fyne is a couple of miles down the road, in Poole. It might be worth taking a taxi there but, if you're going to pay for a short taxi ride anyway, you might also want to consider an Italian restaurant that possibly does a bit of seafood as well. It's Da Vito, which is in the Moortown area of Bournemouth and gets AMAZING reviews:
https://www.tripadvisor.co.uk/Restaurant_Review-g186262-d10478386-Reviews-or60-Da_Vito-Bournemouth_Dorset_England.html
(Annoyingly though, like so many small businesses, it's only got a pathetic Facebook page, rather than a proper website, so you can't check the menu in advance).
Bournemouth does have a couple of decent seafood restaurants though:
The Crab, which is part of the Park Central Hotel, is walkable from East Overcliff Drive, as it's just beyond the pier, on Exeter Road:
http://www.crabatbournemouth.com
Reviews here:
https://www.tripadvisor.co.uk/Restaurant_Review-g186262-d1854192-Reviews-The_Crab_at_Bournemouth-Bournemouth_Dorset_England.html
Also just beyond the pier, but right on the seafront, is West Beach
https://west-beach.co.uk
Reviews here:
https://www.tripadvisor.co.uk/Restaurant_Review-g186262-d1005998-Reviews-WestBeach-Bournemouth_Dorset_England.html