5hrs, easy.

3.60 is up to you, OP. There IMO isn't really much worth in Henkaku aside from being able to use a SD card instead of the proprietary Vita nonsense - you can get emulators pretty much anywhere these days, and the OP didn't mention wanting PSP games at all... I could go on, but I won't. He's also getting his from a random guy, so it might as well be on 3.65 already.

As to game recommendations...

and last but not least...

...A Nyko power grip. Has an extra battery about the size of the Vita one (though, your charging port is almost guaranteed to fail from what I read - mine is still fine). It feels a lot like a proper PS controller, which is nice for action games. There are cheaper options, but this is a solid one anyway. Make sure you buy the correct one, as the one for the old OLED Vita is not the same as for the slim... the Old one is overpriced to high heaven as well.

https://www.amazon.com/Nyko-Power-Grip-Vita-PCH-2000-playstation/dp/B00G9X4YRM

Aww. I was just reading through this and thinking "we're having such a nice thread, people are making suggestions and there's no console wars rubbish; no belittling Vita and no forcing Switch viewpoint" and then you go and post this. Like clockwork there's always someone showing up with this kinda stuff every time anyone even mentions Vita these days. Ah well.

I dunno about Switch length, but I can say that my LCD lasts MUCH longer than 3 hours lol. I have an OLED and LCD, the OLED is probably in the 3 hour ballpark you're right; but LCD is like pushing 6 hours - and even longer if you're playing PSP/PS1 games.

Since these are your criteria (and you haven't outright said "not on PC or PS3", here are a few suggestions. First up, RPG's

- Ar no Surge Plus
- Any Atelier game (my favourites are Totori & Meruru)
- Demon Gaze (I'd try this one before Operation Abyss, but if you are after that game, I wrote a review here)
- The Hyperdimension Neptunia Re;bith games (I wrote about the first one here)
- Legend of Heroes: Trails of Cold Steel (review here if you're interested)
- Lost Dimension (review here if you're interested)
- Oreshika: Tainted Bloodlines
- Soul Sacrifice Delta (easily the best hunting game on Vita)
- Tales of Hearts R (review here if you're interested)
- Ys: Memories of Celceta

A couple of the above games have PS3 versions, but most of them have additional content and/or changes on Vita that make them worth buying for the handheld over home console.

In terms of third-person shooters, your choice is vastly more limited, but I'd still suggest:

- Earth Defence Force 2/2017 (review of the latter here if you're interested)
- Mobile Suit Gundam: Extreme Vs. Force (review here if you're interested - it's not solely a TPS but features those mechanics)
- Uncharted: Golden Abyss
- Unit 13

You'll also have access to a back-catalog of PSP & PS1 games that I'd recommend sifting through, but if you already played them on those respective consoles then I suppose that loses some of the luster.

Ha, thanks for the shout-out.

Yeah "Demon Gaze" is worth a recammend & "Demon Gaze II" game is comeing november 2017.

Id also say get all the "Atelier" games you can get your hands on for it.

Also"sword art online", "Digimon Story: Cyber Sleuth", "world of final fantasy", "Tokyo Xanadu".

Some great or good games are Wipeout 2048, Severed, Tokyo Xanadu, Virtue's Last Reward, and Ys VIII.

The system is also good as a one-stop shop for PS1 and PSP games.

If like me you have been adding every Vita game on the psn plus monthly games you'll find you have built up a good library of Vita games for when you go outside and streaming ps4 games when at home is also great.

Certainly not an expensive option. A huge amount of content is available for me without spending a penny. I have 2 Vita's one that I've had for years and another I bought recently which worked out at about £20 after I sold the Sonic Racing game that was bundled with it. It had a fault which is it won't read cartridges but only use it for my US psn account and can use cartridges with my original Vita. Both have the glorious OLED display.

If your just streaming PS4 games battery life can be very good despite the heavy use of wifi and of course technically the games can be of a far higher quality than Switch when streamed.

The proprietary card is still very annoying though. I'd like to get a bigger card than my current 16GB in my main Vita and 4GB in my 2nd Vita but resent the pricing and would rather just juggle games to those I want to play at the moment. It's not worth me investing any more in the Vita anyway. Not that I have invested much in the first place. Probably both Vita's, a few cartridge games and the purchase of the 16GB card all cost about £150 approx. I'm also finding a shift towards using my Linx Vision console.

That is honestly not a good recommendation at all with it coming to PS4/PC in an improved version...

All of those but Severed have come to PS4 in improved versions.

Vita means no buy

I would reconsider, but if you insist, I think adding SEVEN, Memories of Celceta and even the new Lacrimosa of Dana, all YS games, should be a priority. They are far more solid than Persona and Tales of Hearts. Also, avoid Neptunia unless you like shovelware. 

And naturally add the FF remasters if you don't have them. Download every KZ you find as well so you don't suffer console degradation... 

You should review Celceta next, but perhaps someone did review it and I just didn't see it.