First of all,which meat are you talking about?
Was it offered to you from the temple itself as prasadam?If so,then by consuming it you will become pure as opposed to impure.
If not,then please read on the next few lines:
It is clearly stated in our Shastras that without proper Suchi(cleaniness), if a Punya Karma is done, then that karma bears no fruit.In your case the Punya karma is visting temples and offering prayers and worship .
Souche yatnah sadaa Karyah souchamulo Dwijah smritah|
Souchaacharavihinasya Samasta Nishphalah Kriyah||
Meaning-Every Dwija's(Brahmin,Kshatriya & Vaisya) base(mula) rests in
suchi(cleaniness).So ,they should always take care to maintain
it.Because, works/rituals(kriya) bear no
fruit(nisphala ) if done without suchi(Souchaacharvihinasya).-Daksha Smriti & others.
A human being becomes impure in each of the following cases:
1)On waking up from sleep.
2)On evacuating bowels.
3)On touching dead bodies,attending funerals etc.
4)On having sexual intercourse.
5)On shaving,cutting hair,nails.
6)After applying oils on body.
7)After eating meat,drinking alcohol.
So,it is mandatory, that you bath. if after having done one of those above,you want to participate in a religious ritual.
Duswapnam yadi pashettu vaante vaa khurakarmani, Maithuney preta dhume
cha snaanameva vidheetate/ ( In the cases of viewing disturbing
dreams, after vomittings for whatever reasons, after shaving and
haircuts, after the act of feminine congress, and nearing and touching
a dead body, one ought to purify by Snaana(Parashara Smriti).
Offences against the Deity (bhagavat aparādha) These all relate
to misconduct in a temple or shrine or inattention during ceremonial
worship of an Icon. They are as follows;
Approaching the Deity without brushing the teeth.
Entering the temple or shrine after touching a menstruating woman.
Approaching a sacred precinct after visiting the cemetery.
Entering a temple or shrine wrapped in a blanket.
Weeping or lamenting in the temple.
Combing the hair, or clipping nails or spitting in the vicinity of the temple.
Carrying an umbrella while circumambulating the temple.
Sleeping or eating in the temple or shrine.
Seeing a corpse and thereafter touching the Icon without having washed the hands.
Touching the Icon after having touched a corpse without having bathed.
Touching a sacred icon after having sex (without bathing).
Sitting upon an elevated seat in front of the Deity.
13. Eating meat prior to worship