If you want transparency in IR region, band gap of the semiconductor should be equal to or more than the energy of photons used for transmission. This condition is same for direct and indirect band gap semiconductors. However, the absorption coefficient is more in case of direct band gap semiconductors as compared to indirect one. It has a meaning that direct band gap materials will absorb the light of higher energy than band gap in a thin sample while indirect band gap material of the same thickness will not absorb completely and part of it will be transmitted.
On my opinion the best infrared transmitting materials are chalcogenide glassy semiconductors (ChGS). The range of transparency ChGS is changing from 0.7 mkm to 10 mkm (As-S system), wider range is in As-Se system (1-12 mkm) and in As-Te system ( 2-14 mkm). In general the refractive index in IR for such materials is vary from 2 to 5. See pioneering publication: Rudolf Frerichs, New optical glasses with good transparencyin the infrared, J. Opt . Soc. Amer. 43, 1153(l953) .
The level of transparency (T) in IR for low absorption(α) in samples is approximately equal: T=2n/n2+ 1, α>1, T=(1-R2)e-αd.
In low energy side the transparency is limited by multiphonon absorption. See details in books: Mott N.F., Davis E.A. Electronic Processes in Non-Crystalline Materials.Oxford: Clarendon Press, 1979; Kokorina, V.F., Glasses for infrared optics. 1996: CRC Press.Hilton, A.R., Chalcogenide glasses for infrared optics. 2010: MacGraw Hill. Publication for example :D. Lezal, ―Chalcogenide glasses–survey and progress‖, J.
Optoelectron. Adv. Mater., vol. 5, no. 1, pp. 23-34, 2003. Savage, J.A., P.J. Webber and A.M. Pitt, The potential of Ge-As-Se-Te glasses as 3–5 μm and 8–12 μm infrared optical materials. Infrared Physics, 1980. 20(5): p. 313-320. 960. Hilton, A.R., D.J. Hayes and M.D. Rechtin, Infrared absorption of some high-purity chalcogenide glasses. Journal of Non-Crystalline Solids, 1975. 17(3): p. 319-38.
For crystalline Ge in IR n=4 and the transparency is 47%. Similarly as in ChGS in high energy side for crystalline materials the transparency is limited by interband transitions. The types of rules which might govern of this processes you mentioned in your question. Only level of absorption is limited application IR transmitting materials.