Powering an iComSat GSM Shield and a Gear Motor with LiPo Battery
The GSM shield looks to pass the input directly to a LDO, and regulate it down to 3.8V? I’ve no idea why they claim such a high input voltage requirement, I could be missing something.

With care you could patch the VIN &amp; VBAT leads together on the module, and power them both from 3.8V. The LDO should drop itself out of the circuit, although check it’s not getting hot (e.g. dissipating any power). That device doesn’t explicitly state it can be back-powered like some LDOs will, but if you tie VIN &amp; VOUT together it should prevent damage.

No promises though, if you break anything taking my advice I’m not liable!

But this would leave you with all logic powered @ a lower voltage.