A cheap day return Luton-Lymington Pier is £24.55 with a Network Railcard. Outward travel is valid on any train after 09:30, return travel is valid on any train.
Luton is a related station to Bedford, Watford Jn and West Hampstead.
A Bedford-Lymington cheap day return is £25.10, therefore Bedford is not an appropriate routing point.
A Watford Jn-Lymington cheap day return is £27.90, therefore Watford Jn is not an appropriate routing point.
A W Hampstead-Lymington cheap day return is £23.70, therefore W Hapstead is an appropriate routing point.
Lymington Pier is a related station to Southampton and Weymouth.
A Luton-Southampton cheap day return is £22.50, therefore Southampton is an appropriate routing point.
A Luton-Weymouth cheap day return is £35.80, therefore Weymouth is not an appropriate routing point.
West Hampstead-Southampton has routing permissions London, CW+LB, TS+WX.
CW+LB gives:
West Hampstead-Blackfriers-Herne Hill-Tulse Hill-Croydon-Three Bridges-(Lewes optional)-Brighton-Havant-Portsmouth-Southampton
West Hampstead-Blackfriers-London ridge-Tulse Hill-Croydon-Three Bridges-(Lewes optional)-Brighton-Havant-Portsmouth-Southampton
West Hampstead-Blackfriers-London Bridge-Forest Gate-Croydon-Three Bridges-(Lewes optional)-Brighton-Havant-Portsmouth-Southampton
TS+WX gives
West Hampstead-Willesden Jn-Richmond-Clapham Jn-Wimbledon-Woking-Basingstoke-Southampton
West Hampstead-Highbury-Stratford-Liverpool St-Waterloo-Clapham Jn-Wimbledon-Woking-Basingstoke-Southampton
West Hampstead-Highbury-Stratford-Barking-Fenchurch St-Waterloo-Clapham Jn-Wimbledon-Woking-Basingstoke-Southampton
West Hampstead-Gospel Oak-Barking-Fenchurch St-Waterloo-Clapham Jn-Wimbledon-Woking-Basingstoke-Southampton
West Hampstead-London has routing permissions LB and TS
TS contains the routes to London described in TS+WX
LB contains
West Hampstead-St Pancras-Thameslink Core
London-Southampton has routing permissions CW+LB, CW+PD, CW+PD+SC, CW+SC, MW+WR, MW+WX, WX
CW+LB has already been described.
CW+PD gives:
Waterloo/Victoria-Clapham Jn-Wimbledon-(via Woking or Oxshott & Effingham Jn) Guildford-Havant-Portsmouth-Southampton
CW+PD+SC gives:
Various (most, if not all permutations) suburban routes to Epsom, then Leatherhead-Effingham Jn-Guildford-as per CW+PD
CW+SC gives
Various suburban routes to Epsom (not out of Waterloo, however)-Epsom-Dorking-Horsham-Pulborough-Arundel-Ford-Havant-Portsmouth-Southampton
MW+WR gives
Paddington-Reading-Basingstoke-Southampton
MW+WX gives
Waterloo/Victoria-Clapham Jn-Reading-Basingstoke-Southampton
Waterloo/Victoria-Clapham Jn-Wimbledon-Basingstoke-Salisbury-Southampton
WX gives
Waterloo/Victoria-Clapham Jn-Wimbledon-Basingstoke-Southampton
So yes, Mr Cub, your route is indeed valid.
Oracle, doubling back between Fratton and Portsmouth is allowed, as Fratton is included in the Portsmouth group. Use of the avoiding curve is also allowed because Cosham is included in the Portsmouth group as well.