CRJ700 Captain reported a malfunction that precludes flight into icing and resulted in failure of ATC and Company Dispatch to understand crew requests. ATC's failure to expedite escalated flight to a fuel critical situation.

Narrative

Received 'L ENG BLEED' amber caution message on climb out; followed by 'L COWL A/I' amber caution message. I was Pilot Monitoring (PM) for the and directed the First Officer (FO) to continue the climb and maintain aircraft control; while I ran the Quick Reference Card (QRC) and Quick Reference Handbook (QRH) and covered the radio. I ran the L ENG BLEED QRH procedure. The procedure directed us to; 'Leave icing conditions to prevent ice accumulation on inoperative cowl'. As we were then operating in instrument meteorological conditions (IMC); I directed the FO to continue the climb visual meteorological conditions (VMC) which were forecasted at our planned final cruising altitude. We reached VMC conditions and I communicated our situation to ATC and Company Dispatch (including company maintenance through ACARS). I conveyed to each that we would need to descend in VMC conditions to our destination if able. ATC advised us to inform the next Controller and handed us off to the next sector. There was confusion apparent with both ATC and the Company regarding our situation. I attempted to state as clearly as possible that we would require VMC descent from our cruise altitude. We were then given a descent clearance through IMC ahead. I declined the clearance and again made our situation known - that we were unable to enter icing conditions. ATC queried several aircraft on the arrival ahead of us. All reported IMC but negative icing. ATC relayed this to us and; I believe; again issued a descent clearance. When I declined again; ATC informed us that no aircraft had reported icing in the descent. I stated that we would be unable to accept a clearance into any IMC conditions as that would put us into conditions conducive to ice formation as defined by our Pilot Operating Handbook; and that the negative reports of the aircraft ahead were inconsequential to our current condition. I asked for a hold at altitude so that we could consult with the Company and asked that ATC offer assistance by checking other potential ZZZ2 arrivals or a radar vectored descent to ZZZ2 or any nearby airport with VMC conditions from our altitude to the surface. ATC said that they; '...cannot see clouds;' but that they would check. We were assigned; and entered; a hold at ZZZZZ1; where I conferred with our Company Dispatcher who checked weather at ZZZ3 and ZZZ as requested; however apparently also misunderstood the situation and suggested that we were okay to descend through IMC; after having consulted with a Supervisor. The FO and I separately calculated bingo fuel and were within 100 pounds of each other. I informed the dispatcher of remaining fuel on board (FOB) and our time remaining until bingo holding fuel. As we neared our bingo fuel the FO and I made the decision to divert to ZZZ as weather conditions were most clear there and our descent would put us through the least amount of IMC within our range. We requested vectors to ZZZ and [requested priority handling] due to the required entry and descent through IMC/icing conditions. I also informed ATC of minimum fuel status at this time. We were vectored and subsequently assigned the ZZZZZ 3 arrival into ZZZ. I informed Company Dispatch that we were diverting to ZZZ; had [requested priority handling]; and stated our FOB. We descended through IMC into VMC conditions as I finalized programming the FMS and approach. ATC requested a statement of flight conditions. I relayed that we were now VMC. ATC asked if I was cancelling the [priority handling]. During these transmissions I showed the FO our updated fuel numbers with the arrival and long-side approach to ZZZ now built into the FMS. Our new fuel-over-destination (FOD) calculation was 1830lbs; which was below our required planned reserve of 2224lbs. I elected to maintain [priority] aircraft status and relayed this to ATC with an explanation of the new [situation] being [priority] fuel.' We continued the arrival. On arrival; it seemed that several aircraft were being vectored to land ahead of us; and we were tracking away from the airport; albeit on the ZZZZZ 3 arrival. I asked ATC for a turn direct to the airport; was questioned; and I again stated that we were [priority] fuel and do not want to be flying away from the airport. We were then given a vector towards the field and assigned landing Runway XXR. Once I cleaned up the approach in the FMS our fuel numbers were again above required reserve amount. FO landed the aircraft at 2130 FOB."