Tickets are issued specifically to either Lymington Town or Lymington Pier - there isn't a group station here, like there is with London Terminals, Reading/Reading West and so forth.
If you have a Lymington Town station it's therefore not valid to continue onto Lymington Pier. The fact the ticket is routed "Any Permitted" doesn't change the fact you're not allowed to travel beyond the destination shown on your ticket.
Technically speaking you could be treated as ticketless and penalised accordingly in the (admittedly unlikely) event that the guard checked tickets in the short time between the two stops, or that there were Revenue Protection staff onboard between the two stops or at Lymington Pier station.
I have to admit, the sleepy little Lymington branch doesn't exactly strike me as a hotspot of fare evasion that SWR are going to be hot on cracking down on, but to do things correctly you should really ask the guard whether you can continue onto Lymington Pier, or see if you can get a ticket office to issue you with an "overdistance excess".
Ironically, because the fare is the same to both stations, which means it would be a "zero fare excess", this is often harder to obtain as staff sometimes (wrongly or otherwise) believe they will be pulled up for any zero fare excesses they issue.
