With that budget in mind, an LS conversion is definitely out of the picture. But a 5.7 hemi conversion might just slide in under the wire.
An AEV hemi conversion kit is $6,300. Add maybe a $1,000 to that for a few additional parts needed - new power steering pump (this requirement may have gone away....) O2 sensors, other sensors, fluids, etc. $1,000 might be high if the new power steering pump is no longer required. But staying with that, you would be up to $7,300. Can you find a decent 5.7 and 545 trans for $2,700? Probably. The key is getting a COMPLETE take out, still bolted together engine and trans, with all front drive accessories and the motor mounts.
You need to pay attention to what year motors and transmissions from what vehicles will work best with the AEV kit. AEV can help with those issues. Or you could try one of the other hemi swap kits, like JSS. Probably cheaper, but you get what you pay for, IMO.
This of course assumes you do all the labor yourself. If you have a two post lift, taking the body off makes the swap pretty straight forward. Body on, PITA, but doable I guess. (No personal experience with the body on.)