In addition to the fairly well known 'Due Sonate a Liuto solo' (leipzig 1746) I also have a (poor) copy of BL Ms Add 31.698 which probably contains some pieces by Straube since the front fly leaf bears a sticker 'Lute/Rudolph Straube'. Indeed this may be the very 'lude poog' referred to in the satirical anecdote noted on the end fly leaf of this Ms which tells an amusing meeting between the famous English painter Thomas Gainsborough (1727-88), himself an amateur viola da gamba player, and a heavily germanic accented Straube when G bought S's lute book for 20 guineas (a huge sum) but then failed to persuade S to come and teach him the instrument! The exchange ends:
G' ............. Come home with me and give me my first lesson
S I vill gome tomarrow
G You must come now
S I musht dress myshelf
G For what, you are the best figure I've seen today
S Ay musht be shave
G I honour your beard
S Ay musht bud on may wik
G Damn your wig, you cap and beard become you. Do you think if Vandike was to paint you he would let you be shaved'
Wether there is any truth in the anecdote might possibly be doubted but it's certainly a good tale.
All the pieces in the Ms are for the Dm tuning and some are by Falkenhagen (from his opus 1 and 2 collections), and some elementary pieces/lessons might be teaching material (including an amateurish attempt at staff notation) but I guess some of the more extended works might well be by Strauble and are in a similar style to his published sonatas. But then again they might be more pieces by Falkenhagen since no composer is mentioned in the Ms.
I recall Thomas Schall did a modern intabulation of some of the pieces a few years ago which might still be available on his site. But wether these are Straube original compositions is not known.
Incidentally on one page the instrument is called 'Theorboe lute': presumably the theorboed type of german 13 course lute.