Arguably, the rogue may stand up
The character is not literally "granted" a resource that is called movement but that is in fact the way the rules of movement talk about how it is used:
However you're moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving.
In practice, movement is a resource that may be "deducted from" or "spent" (see Being Prone) and which can have "cost" penalties (per Difficult Terrain and crawling in Being Prone). Additionally, the Normal and Skirmisher movements are granted/refreshed using the same terminology:
Normal movement rule: On your turn, you can move a distance up to your speed.
Skirmisher movement rule: You can move up to half your speed as a reaction
So if the two types of movement are obtained/granted using the same "you can move" phrasing and movement is talked about as a resource that can be spent independently of whatever physical space may be traversed (e.g. over difficult terrain), it stands to reason that the movement associated with the Skirmisher feat may be spent to stand up from being prone the same way a character can with normal movement.
In the related question about Booming Blade, the selected answer makes this same point. It is possible to use the movement resource for something other than directly moving. To wit: one "spends" 10 movement to move 5 physical feet through difficult terrain. It is because of this ability to use movement (the implied resource) without moving that Booming Blade does not trigger in response to standing up.
In short, if you subscribe to this interpretation, where the phrase "you can move" is functionally identical to "you are granted movement," and that there is a movement resource that is independent of a physical translation across the battlefield, then standing up is permitted in this situation.