There are reasons to think Proto-BDS had pluralia tantum. However, the languages that have emerged out of it have done some interesting distinct things with them.
0. Pluralia Tantum that go back to PBDS
Although all descendant languages have developed new pluralia tantum since then, a few can be reconstructed as far back as then:
*śigdir - stars
*lixtan - any structure made from spokes
*t'undan - waves
*xajir - itching, spots
*mit'san - freckles, spots
*p'arir - mist, smoke
*t'ik'rir - fur
*t'igdar - a catamaran-style type of boat
Some cultural notes: Proto-BDS thought seems to have thought that every star consists of multiple entities, and that talking about them as agglomerates made the most sense. In Bryatesle, Dairwueh and Sargaĺk stories of encountering a shooting star generally include rather "plural" notions. It is conceivable that the origin goes back to an even earlier verb *śig, signifying 'flicker, flutter'. In this sense, even one star is "the flickers". It is also possible that the Sargalk word t'iśkɨl - butterfly -, the Bryatesle rysih - shake, quake -, and Dairwueh sidzi - flap, slowly fall by sideways motions (like a leaf)- originate with this verb as well.
In Sargaĺk, there are dialectal differences in how these are handled. In southern varieties, just set the number 'one' before them to specify that you are talking about one. The southern variety has originally had the same system as the northern and western varieties, but has simplified it a bit.
dər śixs-air - one stars.
In northern and western varieties, 'one' is further inflected with a plural congruence marker
dəy-air śixs-air : one_s star_s
the example is from a dialect that dissimilates dər-air into dəy-air
With a few other words, such as 'which', demonstratives, etc, there is a double marking: a plural marker followed by a singular marker. Far western dialects, however, just have it be in plural, followed by 'one' in plural, and finally by the word itself.
Eastern Sargaĺk has created singular forms for most non-pairwise pluralia tantum, and for the pairwise ones, "pair" - mihyor - is the singularizer. There is one further exception to this, lixtan's reflex yuśtan, which has the singularizer miśrik. A few words retain their plural morpheme as part of the root.
Dairwueh has some lexical quirks in the use of adjective and verbal congruence, and may demand normally singular adjective stems with plural markers for these nouns, and the same holds for verbs. Non-nominative cases for some pluralia tantum are singularia tantum instead, and some speakers prefer to use singular congruence markers for these as well. For some speakers, congruence can be used to distinguish a singular referent from a plural referent.
Standard Bryatesle uses counters to turn them into singulars; in many ways, they resemble mass nouns in Bryatesle, and in fact, the plural morphemes sometimes appear on new mass nouns. The syntactical differences between apparently plural mass nouns and pluralia tantum are that mass nouns always take some type of counter-like noun to enable numbers or certain other quantifiers, PTs do not require that for numbers larger than one and PTs always take plural congruence on verbs regardless of actual number, mass nouns always take singular congruence.