I don't know about the weight thing but I could comment on your motor. First and foremost is your car equipped with VTEC? I have a 92 Hatch Si and it has VTEC. If so then kudos, swaping in a VTEC motor will be easy and, in my opinion, the best for running high RPM applications. Personally I would not want the B20 for an all motor build because of the lower compression ratio. Unless you plan on boosting that baby or spending a ton of money on raising a perfectly good B20's already boostable compression ratio I would stay away from it. You can LS/VTEC swap the head's and gain some valuable rev's but still, the comp is way low. The Type R is the ideal candidate for a high rpm, fast Civic but as you mentioned the price tag is uber high. Plus you will have to deal with swaping from OBD I (your year) to OBD II (the Type R motor). My advice, go with something like a GSR or maybe even a Civic Type R motor. The Civic Type R will make around 185 horses to the crank which is nothing to laugh at in a B16. And you can get the full swap package from hmotorsonline.com for $3800 which includes the rare S4C tranny equipped with factory LSD!! Just depends on how much you are willing to spend. You could also go with a hugh H series swap and be out of this world fast but that too is a complicated swap.
Personally, I have a B16A2 sourced from a 92-95 SiR II Civic headed my way and it is supposed to make 170 hp at the crank. With a $1700 Buddy Club Spec IV head job I should be making somewhere in the ballpark of 225 to the wheels. The motor only cost me $2250 and when you figure in the new head job you are looking at under $4000 for an SiR II motor and a fully rebuilt head (cams, gears, valves, springs, retainers) that is cheaper than the Type R and should hang with most things out there. Like I said bro... just depends on how deep them pockets are.