As far as I remember, qbasic.exe is not one of those "self extracting" EXEs.
Back to the origional topic, while I have no idea on why the EXEs don't work, you may add a SHELL command plus setting COMSPEC environment variable at the autoexec.bat of your booting floppy to try make it use the DOS on harddisk after booted successfully.
Note: I'm assumming your harddisk is formatted to FAT16, not FAT32, and your harddisk is not bigger than 128GB. If your harddisk is bigger than 128GB, you may have to adjust the jumpers on your IDE harddisk to place a cap on the size being detected.
I think there could also be issue regarding the size of memory, etc. But since you mentioned those applications run fine if booted from floppy, let's neglect them for now.