Can a deployable ILS operate in CAT II or CAT III?
This article only states Cat 1.  Deployable ILS project has been ongoing for 7+ years, so -1 to you for not posting anything relevant about it in your question.

The D-ILS is to provide a system of equal performance to existing Category I fixed-based systems that provide aircraft guidance on final approach in low-visibility and low-ceiling weather conditions.

https://www.militaryaerospace.com/articles/print/volume-27/issue-11/rf-microwave/thales-developing-deployable-instrument-landing-system-for-air-force.html
March 2010 article on expected May 2010 Request For Proposal release:

The final RFP is anticipated to be out in May with source selection in June, and an award by December. Initial operational capability is scheduled to occur in early 2013.

http://www.aero-news.net/index.cfm?do=main.textpost&amp;id=c873f7b8-2dd6-408f-9da2-b85f5b36a7ea
Here is another that says it is available in a transportable configuration and says it can do Cat II.  Is it available?  I don't know, the AF seems to be buying the Thales version currently.

Transportable and permanent configurations
Meets icao cat II ILS specifications
For Ground Controlled Approach (gca) operations, the ILS provides
controllers with par-like displays for up to four aircraft. All ILS glide slope
and localizer signal parameters comply with icao Annex 10 standards
for Category II ils, including accuracy, integrity, displacement sensitivity,
clearance, alignment, modulation and frequency. The fixed-based ILS has
been approved by the faa for use in the National Airspace System.

http://www.anpc.com/wp-content/uploads/2011/08/ILS-ANPC.pdf