Gear oil, would be a terrible choice. Synchromesh may help, but it's a temporary thing, any fluid will be. Eventually the trans will need to rebuilt, and personally I see no point in replacing it with another used transmission. If you were to go that route though, the shortest geared cable trans is a YS1 from a 92/93 Integra GS-R or 92/93 JDM Integra XSi. LSD was optional on these.

Watch out, EVERY 92/93 cable trans has a YS1 sticker, but most of these are from plain jane B18A Integras with the long gearing. The nice thing though, is that ANY 92/93 YS1 transmission case can accept the newer hydro gears and diffs for a custom gearset of your choice. Switching to a hydro case is a waste imo.