I wouldnt have thought the woofer would have many problems being full range its design probably makes it more sensitive to low frequencies than higher ones!
I woud have some concerns I suppose about running tweeters without any kind of high pass filtering..
If you dont want to put the crossover in there then put a capacitor in series with the tweeters.
C=1/(2 x Pi x R x f) is the formulae used to calculate the capacitor required. R is the tweeter impedance probably 4 ohms and f is the 3dB (half voltage) point.
so to have half the voltage accross the tweeter at 1khz C = 1/(2 x Pi x 4 x 1000) = 400uF approx. But you will have to play with where you want the cut off point!
This type of fitlering gives you 3dB per decacde rolloff ie the voltage accross the speaker at 100Hz is half of what it is at 1kHz and at 10Hz its half of that of 100Hz.
Crossovers normally have a 6dB roll off so 1/4 instead of a half!
I think all of the above is correct but its been a couple of years since I've done any of this stuff

I make mistakes I am only human
hope it helps
Matt