Wouldn't want anyone else to have to suffer through this so here's the log:
There appeared to be two issues here.
1) Spring not properly aligned.
2) Gear splines not aligned.
Due to one or both of these, the transmission would slide forward to about 1.25 inches from mating, then would stop and no amount of wiggling, shaking or tapping would budge it.
Issue 1) Spring slipped out of position during clutch pack alignment. Result: clutch pushrod is obstructed from bottoming out where it rests on the spring, which prevents the transmission from sliding forward into place. To prevent this:
1 -- Before removing clutch pack: -- insert the clutch pushrod as far forward as it will go, pinch it between your thumb and forefinger, pull it out, and measure the distance from your thumb to the tip of the rod. Doesn't have to be exact but it should be in the 2.125 - 2.25" range. This is the 'before' measurement.
2 - Install the clutch pack and make sure the distance the clutch pushrod slides forward matches the 'before' measurement above. If it doesn't, something's not right -- shine a light in the bore and confirm that the spring isn't centered. At this point you'll have to install the clutch pack as many times as it takes to get it aligned right. Note that every time you do it the tooth washers will lose integrity -- can't hurt to have a couple sets of these on hand.
Issue 2 - Splines not aligning
I've installed K75 transmissions six times and have never experienced this to the extreme that the teeth simply won't mesh, but obviously that can happen because it did. If you know the pushrod is bottoming out at the spring, but the transmission still hangs up about 1.25" out, you'll have to get the teeth to align. Guide pins are a huge help, and jacking up the transmission so the bottom of the case lines up with the bottom of the bell housing makes things easier. If it still stops at about 1.25" out:
1 - Make sure the transmission clears the frame on both sides
2 - Put it in gear, turn the output shaft a small distance as Tim and rbm suggested below, and give it another shove forward until the teeth mesh and the unit slides forward. This may take multiple attempts to get it aligned right. To do this, I just attached the driveshaft to the output shaft and turned the driveshaft. First two times it didn't go, but the third time is slid right in.
Tapping with a block and hammer can help nudge splines that are barely misaligned - that's always worked for me in the past. But if they are completely misaligned as these were then no amount of whacking will force them. I can confirm that when all is aligned properly the unit will slide in with basically no effort at all and all that's left to do is tighten the bolts.
Once that is done, there's the issue of aligning the transmission with the frame so you can screw in the mounting bolts. That can be a challenge since due to the funky angles all around there's nowhere to set a bar to use as a lever. I think once I even tried to lever against the coil mounting bracket. I discovered at that point that 1) the part is plastic and 2) you can get them pretty cheap on evilBay since I destroyed the original by levering off it. Since then, I do this:
1) Put the bolt in whichever bore allows you to use the lever action shown in 2) below on the other side of the frame. What I mean by that is that it's easier to use leverage at this point to shift the frame upwards than it is to shift it downwards. If the frame is too high on one side, maybe it's best to remove both bolts, then put a screwdriver in the bore on the side that is higher and try to shift it downwards, so that the other side is lower. You should be able to line up one side pretty easily by just sticking a Phillips screwdriver in there as long as there's no bolt in the other side.
2) Cover a heavy duty screwdriver or pry bar with a cloth at the point shown in the picture and use leverage to push the frame upwards in relation to the bore in the transmission.
From this lever position you should be able to get enough adjustment to get the mounting bolts in.
Hope this helps -- if you have other tips or corrections feel free to share them.