A regular spray bottle will work fine as long as you use a pure liquid mop (i.e. without rub added). If you do add rub to your mop, try putting a piece of cheesecloth over the bottom of the spray tube using a rubberband to secure it.
One thing I've done that I think works a lot better is I have a small (1/2 gallon) quality "bug" sprayer that I use. The long nozzle allows me to get closer to the meat surface through the vents and you also can adjust the spray a lot better. Just make sure this is ALL you use this sprayer for and clean it out with hot water after you use it to keep it from getting sticky and to keep sweet loving pests such as ants away when you store it up. The small container size also fits nicely in the fridge but the hose is a little awkward.
Good luck,
Budman