Haha! It would be unconventional to use a double s in any form aside from the s-perfect, which is the time form of a verb, the superlative forms of adjective, or beyond the first two syllables. After a bit of research an accurate word would be either a g, k, or a c (all formed in approx. the same area of the mouth as x) instead of the x followed by the -ēs. Trying to pronounce Kleenex with an i before the g, k or c sounds less like Kleenex than index sounds similar to indices. The transition from a vibrating sound like the n to an i also feels unnatural at that part of a word, normally it comes after a t or a d sound. Also the name itself stems from the verb "to clean" and the latin suffix -ex. If it suggests something like a "cleaning king" the resulting word could therefore be Kleenegēs, but Kleenecēs is not off the table.

However, my initial suggestion is inaccurate. The senex-style plural (Kleenēs) is a special case of which it ism't resolved why senex drops its -x entirely on flexation.