According to wikipedia,
Weiss probably wrote more than 1000 pieces for lute, from which about 850 attributed pieces survived, most of them grouped into 'sonatas' (not to be confused with the later classical sonata, based on sonata form) or suites, which consist mostly of baroque dance pieces. Weiss also wrote chamber pieces and concertos, but only the solo parts have survived for most of them.
According to Sylvius Leopold Weiss : The Complete London Manuscript the chaconne you are referring to is from "Duo No. 4 in g minor SW14". Accordingly, the chaconne itself was originally written as a duo, not a solo piece. It is subsequent generations of guitarists and lutenists that have adapted the chaconne as a solo piece.
- Chaconne and Duo 4 in g minor WeissSW 14 : both lutenists and guitarists have been playing this chaconne for years as a
solo. Indeed, if we vary the chordal sections with arpeggios and ornamentation, the result can be very satisfactory as a solo, which
brings me to reiterate the possibility of Weiss playing such versatile pieces in one format or another. If the piece were played as a solo,
the reprises would be justified. However, it is clear that as a duo, the result is splendidly balanced. The idea that the movements
preceding the chaconne could be played as solos as well is valid since they are adaptable too (the minuet whose 15 first bars are used
in the solo minuet p.11 reinforces this idea) but it is clear for me that the lute writing here is for a duo concept: look at the regular
alternation between melodic lines and steady chord progressions.
AllMusic also provides details of the available recording of The London Manuscript, so the full sonata must be extant.
The difficulty of locating a full manuscript of the sonata may be due to its being a duo rather than a solo. The popularity of the solo version of the chaconne makes it easy to obtain.