Heading through Southampton city centre to Lowford via Itchen bridge isn't pleasant a lot of the time, you will get a lot of motor traffic passing you. Bridge Rd is part of the A27 which can be horribly busy, I've only ridden that section of your route to literally over the bridge and turned left into Swanwick Lane, which can be a fairly pleasant route outside rush hours to head towards Botley or Curdridge.
https://www.strava.com/routes/2966768861350255228 is a slight variation from my commute home, following my route from Cemetry Rd to over Cobden Bridge, far far quieter at 1300-1500.
Heading in a more direct east route means riding through Hedge End and Botley for me from Bitterne SO18, but I avoid that way unless it's well away from rush hours.
It's far more pleasant to head out via Fair Oak, Horton Heath or Durley. Less traffic, more rural feel.
This section from my ride last Friday is a nice ride to the Old Winchester Hill plateau
https://www.strava.com/activities/7212786595/analysis/571/4198 , but you could avoid the hill by heading through West Meon and East Meon from Warnford (which I did part of the reverse later in that ride).
Bat 'n' Ball pub near Clanfield did have an outsde tap you can top up water with before Covid, not tried to use it in last two years.
Your route idea is a long day in the saddle, without meandering into more rural, less traffic lanes!