First off, credit where credit's due: I don't have all the answers. A lot, if not most, of what I say is stuff I've learned at GTA, not firsthand. The people at GTA collectively have the answers. I've just read through the forums a LOT and try to answer a question from what I remember. You should give your thanks to the GTA family, as should I. So basically I just try to give a condensed version of what I've learned from others, but then inevitably screw it up and write some horribly verbose essay of a post (see below for an example lol):
A quick note on the tuned 34's: when you see one that's shooting around 700 fps in .22 that generally means it's been tuned using Maccari parts, which drop the power significantly in the '34. A 34 tuned with stock parts (which is still a pretty darn nice gun!) will be doing upper 700's, generally around 17-18 fpe. Maccari parts drop it to around 15 fpe.
Well, mid 850ish+ fps in .22 with a mid weight pellet really narrow the field down. That kind of power means the supermagnum springers, and you're going to be hard pressed to find a light one simply by virtue of the large powerplant. Not to mention a light supermagnum is going to mean even worse recoil issues than usual. So I think we're pretty much talking 8 lbs bare minimum.
The supermagnum springers that I can think of right now are the Webley Patriot/Beeman Kodiak, the Theoben Elminator/Beeman Crow Magnum, Walther Falcon Hunter/Talon Magnum, Diana 350 family and clones, Gamo 1250 Hurricane, derivatives, and clones, and possibly the Beeman RX series/HW90.
Right off the bat your price constraints eliminate the Webley Patriot/Beeman Kodiak, Theoben Elminator/Beeman Crow Magnum, and the Beeman RX series/HW90; even if you could find one of those used it would still probably be too pricey.
The Gamo 1250/Hunter Extreme/etc are too pricey new. But you might be able to find a used one in your price range. The Benjamin Super Streak is apparently (?) a clone or semi-clone of the 1250 powerplant, and does fall in your price range (off the top of my head about $250-260 at Airguns of Arizona, much more at Pyramyd). The plus of the 1250 family and Benji SS is that they're very powerful, can get nearly 30 fpe in some cases. The downside is that the Benjamin Super Streak is gaining somewhat of an unfavorable reputation for quality control and longevity, but you might be hard pressed to find someone to tune them. And from what I hear parts support for them is poor to nonexistent. But Ed (aka shadow) has an old Brit-made 1250 that he loves. The Gamo Hunter Extreme is listed at 9 lbs, while the Benjamin Super Streak is listed at 8.5 lbs.
The Walthers are also very powerful (26-27 fpe in .22 if I recall correctly), but have had some quality control issues with the internals and the stocks breaking. Gene won't tune them anymore. But some seem better than others, and if you're willing to do a home tune it sounds like they can be made into serviceable guns. The price is ok on them though, especially a used one. The Falcon Hunter is listed at 8.25 lbs as is the Talon Magnum.
The Diana 350 family and clone (B28A) are probably the most popular of the 3 sets of guns in your price range. They're not quite as powerful as the Walthers and the Gamo 1250/Extreme/Benji SS, generally coming out around 24 fpe. But that power level does meet your velocity requirement (24 fpe will push a 14.3 gr CP at 870 fps). Of the three sets of guns they're probably also the ones least in need of a tune out of the box, but people do say the difference is still dramatic. The Diana 350 magnum is listed at 8.5 lbs, the 350 Panther at 8.2 lbs, and the 350 Feuerkraft at 8.0 lbs. My B28A weighs in at 8.5 lbs. The parts support for the 350 family is going to be the best and with it the best and widest availability of tuning.
Of course, all the above are going to be fairly hard to shoot because of the recoil, with the Walthers and Gamo 1250s worse than the 350 family due to the increased power. They're all fairly long guns too, and as noted, none of them are terribly light.
Okay, I feel dirty for saying this, but here it is: there is a way to get a gun that weighs 5-6 lbs and has decent power in .22. Well, you have to be willing to take a slight power drop to around 22 fpe (not exactly a weakling number though!), but you also get way less recoil. It's the Benjamin Discovery. Yep, a PCP. The Dark Side. Frankly, if weight's your first concern, then it blows the springers out of the water. And it should be easier to get good groups with too. Just throwing it out there.
I will note with some dismay that only one of the rifle I mentioned above has a synthetic stock, the Diana 350 Panther. You were right, the good guns tend to be wood. But if you want to you could take the route some people have, and spray the stock down with truck bedliner to give it some protection from weather and minor dings and marks, and also generally a better grip.
Personally, I'd go with something in the 350 family. Also personally, I see two ideal routes within that option:
1) get a B28A and send it off for a full tune, and maybe chop the barrel back and slap on a brake for cocking leverage. An all out tune with Maccari parts would result in a very nice shooting gun, but probably at the cost of a little power, maybe 2 fpe (would result in a CP with a muzzle speed of around 830 fps). Yeah, I know, the power loss doesn't seem very palatable, but I've never heard of anyone who wants to put the stock spring back in the gun following a tune.
2) Get a Diana 350 Feuerkraft Pro Compact and home tune it, keeping the stock internals. This should result in a pretty nice shooting gun, not quite so nice as one with the spring etc replaced, but one keeping full power. It has the added advantage of being shorter, reducing the lock time and helping the balance in spite of the weight (8.5 lbs).
The above is all just my opinions though, and you know what they say about those

And I'm probably biased toward the 350 family because I have a B28.
By the, feel to call me Eric