Pilot reported a malfunctioning fuel gauge and a diversion to a nearby airport after they were unable to complete an approach to their destination airport.

Narrative

During preflight planning; I am certain I confirmed visibility conditions before and after my ETA. However; after passing ZZZ; I unexpectedly encountered a widespread and intense cloud cover stretching for miles in all directions. This was not anticipated based on my flight planning.Compounding the situation; my fuel gauge fluctuated unpredictably; showing readings between mid and quarter at three separate instances during the flight. When the erratic fuel indications occurred again; I became uncertain whether I had the fuel reserves I originally calculated. This uncertainty led to panic; as I could not confidently determine how much usable fuel remained. Recognizing the urgency; I requested an ILS approach to Runway XX at ZZZ1. I was seriously concerned that I was low on fuel and needed to land immediately. ATC began vectoring me for the approach. However; between ZZZZZ and ZZZZZ1; my autopilot unexpectedly disengaged. This led to an unusual attitude; requiring immediate corrective action to level the wings and regain altitude. After recovery; I climbed to 3;000 feet; turned to a 180 degree heading; and initiated a go-around per ATC instructions.ATC then inquired whether I was IFR rated. At that point; I stated that I was requesting a practice ILS approach due to fuel concerns. This response misrepresented the emergency; as I was in fact experiencing an in-flight fuel emergency without an IFR clearance; violating FAR 91.13 and FAR 91.173 (IFR flight requirements). I would not have created the violation were I not of the belief that I was experiencing an inflight emergency. ATC identified ZZZ2 as the nearest airport with VFR conditions; and I proceeded there. Upon landing; I reported the incident.