DA40 pilot reported entering icing conditions without activating pitot heat. In the ensuing high workload; the airspeed indicator was lost; an altitude deviation occurred; and the pilot regained control and safely diverted to an alternate airport.

Narrative

Most of my experience is in a Bonanza B36TC. I am now flying a club rental DA40. The aircraft is shared; so the avionics settings are by others. One in particular is the OAT in degrees F.I have flown in a reasonable number of mildly challenging situations; often with instructors or experienced pilots.I was the PIC and PF. The AP (Autopilot) was reported INOP before my outbound flight. The aircraft has a fuel pressure reporting anomaly: it alerts low fuel pressure repeatedly: a warning chime.I checked WX and planned the flight. I reviewed a full WX briefing ~5 hrs prior to the flight. I re-scanned the WX 40 mins before the flight. I read the icing AIRMET and PIREPS. I also checked METARs along the route. I was aware of MEAs along the route.WX was OVC and light rain; no TS. The route included crossing [a mountain range]. I had flown in less than 48 hrs earlier. I conducted a cursory preflight (inspect for ramp damage; sump fuel; free control surfaces). I got clearance and entered the route in the avionics; departed and expected to enter hard IFR at ~3000 and climb on top around 7000. I entered the clouds as expected; but did not see signs consistent with a ~3-4000 layer. I asked ATC for tops reports; and expected 11000.I got between layers ~7000 with only trace ice. I naturally expected I would climb to 12000 and be on top quickly; from my BE36 experience. I asked ATC for deviations and areas of light precip; and turned N while continuing the climb. I noticed my workload was high.The side windows were obscured with condensation; and I was hand flying in hard IMC. I was not comfortable looking out at the wing root for ice. The climb rate was far slower than I had imagined. I was in the stall margin. The low fuel pressure chime sounded. I observed low fuel flow around 7 GPM; should have been over 10.. I switched on the aux fuel pump; which had no effect. I opened the alt air; which did not help. a few moments later I felt a mild vibration. My workload was at maximum. I made a radio call; requesting priority handling; and began descending.The controller responded and I turned generally toward the nearest field. My airspeed failed; and I realized I had not turned on pitot heat. I turned it on. Moments later; I noticed high airspeed from the sound of air whistling and carefully applied pressure to arrest the descent. (later; on viewing the tracking data; I now see I descended over 1500 FPM).The propeller ice shed quickly; and the inlet ice dissipated. The vibration ceased and the fuel flow returned to normal. I never saw additional ice; though I'm sure it was there. I landed visually without incident and chose to stay overnight to reassess the trip home the next morning.I am confident my errors included:- Inexperience with the aircraft's capabilities and systems (vs. my prior owned aircraft; which I deeply assisted in an engine change; avionics replacement; and routine MX (Maintenance) vs. this AC).- My failure to create a sufficiently accurate mental picture of the cloud cover; icing potential; terrain limitations; and AC performance. While I was aware of each element; I did not combine the realistic limitations of each; which resulted in an optimistic picture 'climbing steadily to 12000 while popping out on top at 6-8000'.- My error in choosing to continue when the picture was substantially not as I visualized.- My error in judging the workload imposed by hand flying in hard IMC; although I believe that helped when the airspeed went offline; as I was already accustomed to hand flying when the AP would have disengaged unexpectedly.- Not internalizing the OAT in degrees C; so I did not react as I normally would when entering IMC at near freezing temps by turning on pitot heat.Over five years ago; I flew often in an intimately familiar aircraft. Although I am current and trained per FAA requirements; the contrast between my previous flying skills and those I posses today is stark. I need to:- reassess (lower) my self-assessed skill level- train more; both independently (FAA WINGS and similar programs; and instructor-led)- review the AC performance limitations; and 'go out and fly' to experience these limitations safely in DVFR conditions; as I had often done in the other aircraft.My recent experience has been unremarkable flight - simple takeoff; long cruise; simple landing. In my past experience; I had explored emergency descents in a proficiency course; high altitude flight with a safety pilot; flirted with icing in a FIKI (Flight Into Known Icing) equipped plane. I find that I have difficulty taking old experiences off of my flying resume. In summary; yesterday; I flew like I had a lot of recent experience with difficult situations even though that experience is not as readily available; especially in the moment.This was a classic failure of ADM (Aeronautical Decision Making). I'm not sure how the system could be improved. I take this experience seriously; and believe I pressed on when I should have turned back.