It seems to me that 'at arm's length' can be construed in two slightly different ways: 'to keep at a distance' (I want nothing to do with it) or 'to have within reach' (I may have need of it at some point). Below are suggestions for both interpretations.
The first suggestion is:
arcere – to prevent from approaching; to refuse to associate with; to keep off, out
per ministros agere et arcere hostem satis habebat
he was content to act through his subordinates and to keep the enemy
Tacitus, Annals, 12.40
Odi profanum vulgus et arceo
I despise the common crowd and keep it at a distance
Horace, Odes, 3.1.1
However, I suspect your three men may be inclined to pick up those glasses again or, at the very least, are not averse to wining and dining on another occasion, so perhaps arcere is too final. My second suggestion, therefore, is:
promptus – to be at hand; to have ready, at one’s disposal or command; but also to be before the eyes thereby capturing the sense of something at a short distance but still within sight. Most often seen in the phrase in promptu (esse, habere, ponere, etc.) – within easy reach for use; or in full view; in a prominent position.
nam haec ipsa mihi erunt in promptu quae modo audivi
for I shall have the arguments I have just heard ready to hand
Cicero, De Finibus, 2.35
Ad salutem omnia parata sunt et in promptu
Everything conducive to our well-being is prepared and within our reach
Seneca the Younger, Epistles, 119
tabernacula aperta et omnia cara in promptu relicta; argentum
quibusdam locis temere per vias velut obiectum ad praedam vidisse.
the tents were open and all kinds of valuables were left in full sight and ready for the taking; here and there he had seen silver carelessly flung down in the
lanes, as if to tempt a pillager.
Livy, History of Rome, 22.42 - I like how this passage captures the sense of temptation when something is at arm's length.
Another, final suggestion which could maybe keep the 'arm' of 'arm's length' imagery is:
ad manum – close at hand, or as the OLD clarifies, “spatial proximity with additional idea of availability”. However, I have to concede that this more often than not could also simply mean "to hand", as in "in one's hand".
ubi ad manum venisset hostis, tum coortos tota vi gladiis rem gerere
when the enemy had come within reach, then they were to assail them
with all their might, and settle the question with the sword
Livy, History of Rome, 2.30
haec arma habere ad manum
these are the weapons he needs to have within reach
Quintilian, The Orator’s Education, 12.5
incedere inter milites, habere ad manum centuriones
[the women] paraded among the soldiers, they had the centurions at
their beck and call
Tacitus, Annals, 3.33