The main source for the history of hobbits is the prologue to the Lord of the Rings. There are mentions elsewhere, including the Tale of Years, The Shadow of the Past and the Hunt for the Ring. The last of these is in the Unfinished Tales, and its canon status is questionable (never finished, several conflicting versions).
There are no records of hobbits entering Beleriand. However, the prologue notes that
Only the Elves still preserve any records of that vanished time, and
their traditions are concerned almost entirely with their own history,
in which Men appear seldom and Hobbits not at all.
My guess is that they didn't. The prologue makes it clear that hobbits are close relatives of men, and (like men) gradually migrated westwards. However, they seem to have moved west much later than the Edain; the first mention of hobbits in the Tale of Years is TA1050, almost 4,500 years after the destruction of Beleriand.
At various times, hobbits could be found living in Wilderland, the Vale of Anduin, the Angle (in eastern Eriador), Enedwaith (south of Cardolan), Dunland, Bree and (of course) the Shire. Most seem to have gradually migrated towards the Shire, but whether any settlements other than Bree and the Shire remained at the time of LotR is unclear. One version of the Hunt for the Ring states that the Nazgul went to the Vale of Anduin shortly before their mission to the Shire, and found the villages there to be long deserted. Another states that hobbits still lived there but that the Nazgul killed them. Notably, in 'At the Sign of the Prancing Pony' it is stated that
Nowhere else in the world was this peculiar (but excellent)
arrangement to be found.
Here, the 'arrangement' refers to hobbits living alongside men, so if hobbits did still live elsewhere, they did so alone. Finally, the prologue states that
the regions in which the hobbits then lived were doubtless the same as
those in which they still linger: the North-West of the Old World,
east of the Sea.
I'd say that makes a hobbit village south of Gondor pretty unlikely, but I don't think we can be 100% sure.