• Unit 13 - this is a TPS game, not an FPS so keep that in mind before making a purchase. It features no Competitive Multiplayer modes - the entire Multiplayer section of the game is focused on co-operation with other players in fulfilling objectives of the missions, so I suppose you could consider it Skirmsh-based.
• Resistance: Burning Skies - this game is in a lot of ways a budget title, so get it if you can get it cheaply. The Multiplayer section is limited to six maps and three game modes - Death Match, Team Deathmach and Survival, none of those feature A.I opponents, so the game doesn't offer what you're looking for.
• Killzone: Mercenary - this is easily the best FPS on the platform today, it features three Multiplayer modes on six maps, much like Resistance, however aside from the competitive Free-for-All and Team Deatchmatch modes, it offers a Objective-based mode called Warzone in which teams have to fulfil certain on-map Objectives to earn points. Again, no A.I opponents.
• Call of Duty Black Ops Declassified - this game suffers from a similar problem Resistance does in the sense that the Single Player campaign is short, however that's not what you're interested in from what I'm reading. It has the widest variety of Multiplayer game modes, meaning Team Deathmatch, Kill Confirmed (Team Deathmatch plus collecting dogtags), Drop Zone (Capture the Flag), Free-for-All and Team Tactical, which mixes all the previous modes into a playlist. It sports six maps and no A.I opponents.

Now, keep in mind that "Call of Duty - BlackOps: Declassified" as well as "Resistance: Burning Skies" were both poorly received. I personally enjoyed them, however don't expect a full-blown "Resistance" or "Call of Duty" experience - they're very much "pocket" versions of their home console counterparts.

 

In a lot of ways, Unit 13 should serve your purposes very well. It's an affordable and fun title which allows for good interaction online and I had a blast playing it. It's not a massive production, so don't expect too much fireworks, but for what it's worth, it's a good game and it was well-received.

Keep in mind that Borderlands 2 is coming to the PSVita and that game fits your needs perfectly with multiplayer entirely oriented around fighting A.I enemies.

 

I thought you specifically meant Multiplayer Skirmsh (you and other players versus A.I), not just an Offline Deathmatch against bots. Now that you've clarified, let me elaborate.

• Call of Duty - BlackOps: Declassified lets you fight against bots in Hostiles Mode. It also features a Challenge Mode where you shoot targets rather than enemy bots racing against the clock. The main campaign called Operations Mode is divided into specific Operations which you can replay for a better score - it's not your typical story-driven campaign and the specific maps are only loosely connected to each other - they're practically tailored to be short, self-contained experiences that you replay over and over. All three modes are only Single Player, your session cannot be joined by Ad-Hoc or PSN players.
• Unit 13 is an objective-oriented game with the campaign divided into separate missions, so in a lot of ways, a good portion of the main campaign is practically Single Player Skirmshes which can be joined by other players over PSN. Additionally the game features an Assassination Mode where you're supposed to kill a specific enemy target by whatever means necessary, so it gives you a lot of wiggle room in terms of gameplay style.
• Resistance: Burning Skies does not feature an Offline Skirmsh mode, it only offers the standard Campaign or the Online matches against other players.
• KillZone: Mercenary is not a game I own yet, however from what I'm reading, it doesn't have a Skirmsh mode per-se - instead it features a Challenge Mode in which you can revisit Single Player missions with additional objectives, each mission having three Challenges to complete.

From what I'm reading, the games that would best fit your needs would be "Unit 13" and "Call of Duty - BlackOps: Declassified"and while the first can be easily recommended, the second received a lot of bad press.

In my opinion undeservingly so as it does specifically what it was supposed to do, it's a small game you play on-the-go rather than a full-blown "Call of Duty" experience gamers were expecting, but nevertheless the reviews say it's bad, and as I don't want to set you up for disappointment, get the game only if it's cheap or simply have a look at a review online before buying it.

I'm literally between a rock and a hard place here as far as the recommendation is concerned because as far as Offline against bots is concerned, "Call of Duty - BlackOps: Declassifed" is literally exactly what you're looking for... except it's not well-recieved.