Maybe if you weighed your 30-30 barrel, we could guesstimate how much the gun would weigh with 4-6" wacked off.
Muzzle diameter .66, diameter at 16.5"=.76, chopping off 5.5" would save about 9oz.
In factory configuration, my 1999 vintage 30-30 syn weighs 3lb 9.5oz barrel, 2lb 10.5oz action, stock and forearm, total 6 1/4lb with open sights. That's 20oz over your 5lb goal, so any way you slice it the gun is going to weigh over 5 1/2lb unless you turn the barrel down on a lathe. I'd guess the velocity loss at around 200fps if you cut it down to 16+". You may be able to get that back with a 30-30 Ackley reamer, definitely could with a 307 Win reamer.
I'd think a 125gr Nosler BT at 2400fps from the short 30-30 would do fine at 200yards. PCBallistics says 6.3" low at 200 with a 100yd zero, if you sight it in 2" high at 100, it will be dead on at 170 and 2.3" low at 200. That's minute of deer and then some.
A Sierra 30-30 flat nose 150gr at 2200fps would hit 5" low at 200yds from a 2" high zero at 100yds. Entirely workable if you hold a bit high on the shots over 150yds.
If Remington's claimed BC is accurate (0.338), their 30-30 round nose would hit 4" low at 200 if zeroed 2" high at 100 with 2200 muzzle velocity. You could use green box ammo in a short barreled handi if those numbers are good enough for you.