Small aircraft pilot flying reported losing TAWS and other navigational aids while descending into the clouds in solid IMC while landing at the destination airport. As the aircraft broke out of the clouds slightly south of intended position the navigational aids returned. The pilot decided to continue the approach and landing instead of performing a go-around.

Narrative

I was flying as pilot in command on an IFR flight plan with another IFR-rated pilot from ZZZ to Santa Monica SMO where I am based. The plane I was flying is Aircraft X with [manufacturer] avionics; which I am very familiar with and earned my PPL and IFR ratings in Aircraft X airplanes. The ceiling was 1;200 ft. AGL but ASOS was reporting scattered at 800 ft. AGL as well. We were on descent using the RNAV 21 approach and LPV minimums of 440 ft. MSL as well as an autopilot coupled approach. We were in solid IMC around 1;500 ft. MSL when we passed MIPTE; which is 3.3 NM 033 degrees from Runway 21.In solid IMC and a stable approach; the moment we passed through MIPTE we received a TAWS (Terrain Avoidance and Warning System) Not Available" audible annunciation. At this moment we lost synthetic vision on the PFD (Primary Flight Display) and the vertical glide path. Quickly; we realized we could still fly the LNAV minimums to 685 ft. MSL. However; a few seconds later I realized the CDI (Course Deviation Indicator) showed GPS LNAV but there was no HSI (Horizontal Situation Indicator) at all. This didn't make sense so I looked at the autopilot mode area on the PFD area and noticed I was no longer in NAV/APR Mode but was instead in ROL/PIT Mode. I realized this meant I was in fact no longer tracking any reference and was simply holding wings level and pitch; which maintained my descent.This entire situation took 10 seconds and at the end of it I realized it was very dangerous; we were still in IMC. I audibly said to my copilot; 'I think we need to go missed;' and right as my hand went to the throttle we broke through the ceiling and the airport was in front of us. Neither of us wanted to go back into the clouds. We were slightly south of where we were used to and that correlates to being blown south in a ROL Control Mode as the winds were from the northwest. Additionally; our TAWS came back as we broke through the ceiling too. At no time did we get a GPS annunciation or any annunciation indicating actual loss of navigation aid. TAWS was for awareness only and we didn't need it to properly and safely fly the approach. We landed uneventfully.In retrospect I should have gone missed immediately. I did not however; realize I had lost GPS or navigation functionality. The only annunciation was the loss of TAWS which I didn't care about and a clear disappearance of the glide path indicator. For a majority of this situation I believed I had lateral navigation. My familiarity with [manufacturer] avionics is quite high but I was not familiar with warning annunciations for loss of TAWS and GPS. I assumed a loss of navigation references would have a less benign annunciation. Studying the [manufacturer] Pilot's Guide afterwards has led me to realize there are several annunciations where loss of GPS would provide a benign sounding TAWS warning.Looking at SD card data logs from the plane. Quite literally at MIPTE the GPS lost 3DDIFF resolution and went to 3D Mode. This corresponds with a loss of WAAS. This lasted for two seconds before WAAS and 3DDIFF was restored. I suspect this is why I never got a GPS warning as I did not actually use GPS. However; I am still perplexed as to why lateral navigation was not available. My GPS altitude jumped 60 ft. in one second during the anomaly per logs; which could explain the plane's reaction."