Triple post, whew...
I got it figured out. You guys suck for not having figured this out earlier!
Thanks for the ideas though Captain Terror, you did help me along a bit. I'll describe my method here for anyone who might be trying to do the same thing (unlikely as it may be.)
I have two separate doors, both in the exact same place on the map (overlapping, if you will.) One of them, the one that stays open for ten seconds, is initially fully visible, and the other one, the normal one that will only stay open for four seconds, has its render mode set to "texture" and FX amount at 0, so it's invisible to start with. So then I have a trigger_once a little bit in front of the doors, used solely to open the ten second door. When the ten second door closes, I used the "fire on close" field to trigger a multi_manager.
The first thing this multi_manager does is trigger a trigger_relay entity which has "kill target" set to the name of the ten second door, so it deletes it from existence in the map, and it won't get in the way anymore. Next, it triggers an env_render, which targets the four second door, and sets its render mode to "normal", thus making the door fully visible. Since these two things are triggered at exactly the same time, the two doors are essentially switching out for each other.
Now, doors that have anything in their name field are automatically locked, and apparently cannot be unlocked by way of a master. That was a problem for me, because my door had to have a name if it was going to be targeted by an env_render. So, I just used a trigger_multiple to open the four second door, and to make sure it couldn't be opened while the ten second door still existed, I instead set the trigger_multiple to have a master, and used the whole multisource/trigger_relay setup on it, which is eventually triggered by the multi_manager I talked about earlier.
So, this is just about the most complicated func_door I have ever created, and probably ever will create. Needless to say, I would recommend avoiding this setup in any circumstance, but if you absolutely have to...well, now you know it's possible, anyway.
Oh, and one last note... I actually did NOT need to be able to go through the door during the first ten second sequence. I don't even want to think about what would be necessary for that, but it would probably involve a "passable" (non-solid) door and an invisible train. My head's about to explode.