I'm going to offer a leftfield option that doesn't require a passport...
Longleat Center Parcs.
~2 miles from Warminster town centre, that has a train station, usually no problem getting your bike on the direct train from the Portsmouth end.
The hire bikes are decent, but you might miss a road bike or higher end MTB. You can even take the Ridgeback ebike hybrids and Specialized eMTBs off campus, which were £65/75 for Mon-Fri iirc.
Windhill bike park just by Longleat estate, a maze of bridal paths and lovely quiet lanes towards King Alfred's Tower.
Option of ~70+ mile loops to The Mendips for the day, without having to tackle many busy roads.
Bit hillier than western South Downs, you will do well to average less than 75 feet per mile on routes that don't cover the same roads heading out and coming back.
Prices plummet once kids are back at school for Mon-Fri stays and the weather is usually good around mid September.