You are somewhat right to be concerned, but it sounds like it is unavoidable if you are going to submit through a commercial pipeline. However, keep in mind that Illumina likely do some kind of first- and second-strand synthesis as part of their library preparation protocol from RNA anyway.
We have been in a similar situation and found that the NuGEN Ovation kit does pretty well and our results look quite good. I would personally recommend making the libraries yourself and just submitting them to a sequencing core. I understand this can be a little daunting if you haven't done it before, but its really just a case of following simple instructions.