I only know two publications about interdune quaternary lakes with partially carbonate settings: Yang et al. 2010 (Recharge to the inter-dune lakes and Holocene climatic changes in the Badain Jaran Desert, western China. QR 71, 10-19) and Sebag et al. 2013 (Paleohydrological Reconstruction from Late Holocene Records in Interdune Lakes (N’Guigmi, Northern Bank of the Lake Chad, Niger). Open Journal of Geology 3, 143-155). There might be something about pleistocene small lakes/ponds in the Negev (Israel) and, maybe, about the Lake in the Hor Al-Hammar Basin (S Iraq), which is surrounded by dune fields (don't know if that setting fits to "interdune") and has clastics but maybe also carbonates in its setting. Unfortunately, I can not help with examples of miocene age.
Best regards
Johannes
edit: Mason et al. 1997 Holocene history of lacustrine and marsh sediments in a dune-blocked drainage, Southwestern Nebraska Sand Hills, U.S.A. J. Palaeolimn. 17, 67-83. might be also helpful. And, more general, Pigati et al. 2014 Desert wetlands in the geologic record. Earth-Science Reviews 132: 67-81.
-Durand A. et al.(1984). Le Nord-Ouest du lac Tchad au Quaternaire. Paleoecology of Africa , Balkema, 16, 215-243.
For present lakes you can see:
-Maglione G. (1976). Géochimie des évaporites et silicates néoformés en milieu continental confiné. Les dépressions interdunaires du Tchad. Travaux et documents de l'ORSTOM, 50, 395p.
-Carmouze J-P (1976). La régulation hydrochimique du lac Tchad. Travaux et documents de l'ORSTOM, 58, 418p.
Lots on Holocene lakes. It depends on what you really need. For example, if you need information on petrography, there are very, very few. If you need information on isotopes, there are lots, 90% of which is from the Quaternary. Examples from the older geologic record are sparse. Look for our paper,
Parrish, J.T., Hasiotis, S.T., and Chan, M.A., 2017, Carbonate Deposits in the Lower Jurassic Navajo Sandstone, Southern Utah and Northern Arizona: Journal of Sedimentary Research, v. 87, p. 1-23.