Positioning using Global Navigation Satellite Systems (GNSS) typically requires several seconds of continuous signal reception from satellites in Medium Earth Orbit (MEO). This requirement poses challenges for applications where receivers can only capture signals intermittently or operate under constrained power and visibility conditions. In such scenarios, maintaining continuous tracking or reliable line-of-sight to GNSS satellites may be difficult, and conventional GNSS frequencies may also be vulnerable to interference or jamming. Low Earth Orbit (LEO) satellite constellations provide an attractive alternative due to their lower orbital altitudes, which result in higher received signal strengths, as well as their operation across a wide range of spectrum including Mobile-Satellite Service (MSS) and terrestrial L and S bands. These characteristics make LEO signals promising for navigation in challenging environments. This work presents a snapshot-based differential positioning framework that leverages signals from LEO satellites. In the proposed approach, a receiver collects signals for short durations (5-10 seconds) before entering a low-power state, enabling positioning with intermittent observations. Doppler measurements from multiple satellites are combined with a differential measurement model using a fixed reference receiver to mitigate common errors such as satellite clock bias and ephemeris uncertainty. Experimental results demonstrate that the proposed differential Doppler framework operates effectively within the constraints of snapshot-based reception. The method achieves a position error reduction of approximately 47% even when only three satellites are simultaneously visible to both the rover and the reference station.