As a general rule, I think it unwise to attempt, solo, for the first time, the sort of climb that one has no experience with, especially if it carries the potential for a serious accident.
You have winter against you, ice against you, inexperience against you, and possibly an unknown reaction to 4,164 meters against you. Solo? No, not unless your meaning of do not have much alpine experience is an extremely modest understatement.
As for hiring a guide, it is extremely likely that a company will be able to find a guide for you, even if they do not regularly offer tours in the winter. You will be hiring a private guide and it will be more expensive than the summer tour. The reason that the companies do not offer regular tours is probably simply because because there are not enough tourists who want to, or are able to, climb a mountain in winter conditions. However, arranging a guide for a private party or a single hiker is probably something they do fairly often.
Breithorn Ascent says:
This tour is a perfect way to get a sense of high Alpine air for the first time but only for surefooted and experienced mountain hikers accompanied by a mountain guide. The tour includes climbing a glaciated four thousand-metre peak, and this is not to be underestimated
•The overall difficulty of the tour is given as “L” (easy)
•A mountain guide is recommended
•Warning: glacier crevasses
Crampons, climbing harnesses and helmets can be rented in different sport shops in the village
This description and caveat is for a summer tour.