It'd take you longer to pull the engine than to get inside and fix the transmission.
As noted above, your shifter dogs are likely worn out, which happens over time (foot resting on the shifter). As for metal parts floating about the engine, not likely at all. If something big broke loose, the oil filter would catch it.
If you wanted to make sure, pull the header and then the oil pan (not terribly hard to do). What's sitting in the pan will be quite obvious.
If you already had the engine OUT, someone who's done it before, and some spare good parts (ie, spare good transmission, or at least 2nd gear parts-gears, shifter dogs, new bolts, gaskets) could easily get it pulled apart and re-assembled that day.