yes,you can come with your family, if your comapany will process the work visa BEFORE the visit visa is expired.

we did the same. we got visit visa at doha airport , and with in 2 weeks the family visas for me and my baby were issued. btw. we didt even have to go any where like bahrain or where so ever.

this is only my personal experience.
i hope it workes for you and your family.

wish you all the best
cheers
fefee