You could use ClipText with a bit of batch file magic, using another of Horst's programs,
LMOD, which can be used to renumber files. Or just a simple batch,
if exist foo1.bar ren foo1.bar foo2.bar
cliptext to foo1.bar
etc. You could presumably set a Windows shortcut for the batch file. This would work as you're going along. If you can accumulate all the clips you want, and only then export them all in one go, there are other possibilities.
The first one that came to mind involves money. Try the
NoteFrog™ version 2 Customer Preview. Turn on
Options -> Turn ON automatic clipboard capture. Capture your clips. When done,
Stack -> Save Active Stack contents as individual items. If you set, say, "out.txt" as the output file, you will get a file out.txt that contains a summary of all the clips, but also a set of files out0.txt, out1.txt, out3.txt, etc. which will contain each clip in a separate file. This only works with NoteFrog v2 preview, it isn't in current full release 1.9.5. They currently have a special offer of $19.95 for all future upgrades.
That idea made me look a little farther - and it turns out you don't need to pay anything! You can do the same thing with my regular clipboard enhancer, ArsClip. Suggest you try
the 3.2.3 test 8 beta, the one I'm using now (the version 4 betas are early, not everything works yet). This works similarly to the NoteFrog example above, but ArsClip is free, with donation possible. Once you've accumulated your clips, popup ArsClip,
System -> Edit History, and mark the clips you want to export from there. The file names are always PopupClip0001.txt, PopupClip0002.txt, etc. and it looks like you can't change the base name, but you could use your file renamer of choice if you need to do that. It may help to go into Configuration and adjust the number of clips in the popup to avoid having to go into Removed Clips as well as current ones.
Here on DC, mouser's ClipboardHelp+Spell (CHS) can export marked clips, but apparently only to a single CSV file.