Bishop is a 'gentle soul'.
We learn from Burke and Bishop that modern androids (sorry, "artificial people") don't have combat training. He's literally incapable of harming humans and this presumably extends to other life-forms.
BISHOP: Well, that explains it. The A/2's were always a bit twitchy. That could never happen now with out behavioral inhibitors. Impossible for me to harm or, by omission of action, allow to be harmed a human being.
The gun is dead-weight.
In the script and novelisation (based on the script), Bishop is concerned about being slowed down by his equipment. Having to use a free hand to tote around a gun is something he'd want to avoid.
Ripley passed him a small satchel. It contained tools, patch cables and replacement circuit boards, energy bypasses, a service pistol, and a small cutting torch, together with fuel for same. More weight and bulk, but it couldn’t be helped. Better to take a little more time reaching the uplink tower than to arrive short of some necessary item.
Frankly I always viewed his glance and smile at Ripley as basically an "are you kidding me?" look. He's going up against adult xenomorphs. Even if he was capable, shooting one with a pistol is just gonna piss it off.
He can't take the gun
Moving down the canon scale, we're advised in the Aliens: Colonial Marines Technical Manual factbook that not only is it illegal for Bishop-model androids to use weapons but that they're essentially incapable of violent behaviour, serving only in non-combatant roles within the military.
Presumably he would be compelled by his programming to give up a weapon if offered one (assuming his program compels him to follow the law) and wouldn't be able to use it competently even if he chose to violate his programming and pick up a gun
5.1 SYNTHETIC HUMANS (ANDROIDS)
In recent years increasing numbers of synthetic humans have been deployed with Colonial Marine line units. Though the Geneva Convention prohibits the equipping of androids with weapons or uninhibited combat abilities, the Colonial Marine Corps regards them as an invaluable resource, supporting front line units as multi-role team-members and mobile databases. Marine androids are always employed in a non-combatant role, usually as drivers, pilots, medics and scientific advisers to combat units at platoon level and above. Though they are artificial intelligences in the broad sense, legally androids are classified as Corps property and can be ordered to perform hazardous tasks in place of humans? however, their utility and not-inconsiderable unit cost is a disincentive to any Marine field commander who wishes to treat a synthetic as expendable.