Not without making some kind of value judgement about the behavior of the real part for the rest of the frequency spectrum. However, this is not an uncommon approximation to make. If you take a look at various basic dielectric dispersive models (Debye model, Cole-Cole model, ect.) you will find similar approximations. That is, in a Debye model, we assume that the dielectric has a single resonance. The resulting real and imaginary parts are chosen to still satisfy the Kramers-Kronig. Despite such a rudimentary approximation, the resonant behavior is largely confined around the frequency of interest and thus can serve as a limited bandwidth model of actual materials.
That being said, the information beyond your measurement spectrum can still affect the unknown imaginary part of that same spectrum because the Kramers-Kronig relation relates the entire spectrum of the real part to each point in the imaginary part (and vice-versa). So how you choose to model your permittivity outside the measurement can affect your estimated Hilbert transform.
Since you only have a bandwidth that demonstrates normal dispersion, I can see how this could be tricky because you now have to make some kind of guess as to how the refractive index relaxes back down (obviously one must have at least one point where the derivative reverts the sign so that the index will approach the vacuum permittivity again as we expect of materials when we go to infinite frequency).
So probably the simplest thing to do is to allow your real part to smoothly relax back to the vacuum (or general bulk dielectric) permittivity. I can't say whether or not you can do this intelligently on the data that you have. Maybe another thing to do is see if you can't fit one of the single resonance models (like the Debye or Cole-Cole) into your data. The normal dispersion could match up with one side of a resonant permittivity where the resonant frequency is at some frequency above your bandwidth.
EDIT: Personally, I wonder if it wouldn't be easier and more reliable to find a way to estimate the imaginary part simply from measurement. I guess the real part is easy to measure simply via refractory measurements but the lossy part might be crudely estimated from the loss in intensity over the path length in the material. One might rig up a photodetector like a photodiode and use a highly collimated light source like a laser to measure the loss in intensity over the path through the material versus air. One might then obtain a simple approximation to the imaginary part from this.