Air carrier Captain reported on departure receiving wing anti ice messages resulting in the flight crew avoiding areas of icing to the extent possible and returning to the departure airport.

Narrative

During our climb out of ZZZ; the L Wing A/I and R Wing A/I Caution Messages annunciated and indicated to us that our wing ice protection was compromised. ATC gave priority handling aircraft due to the nature of the issue and the conditions of the weather which was reporting ground icing conditions and moderate icing in the descent to all of the airports we could safely reach. Once at FL290; we completed the QRH procedure which had eventually reached the conclusion of 'Leave Icing Conditions'. At that point; we decided as a crew that it would not be wise to continue to ZZZ1 since it was forecasted to snow and instead had ATC temporarily hold us at ZZZ2 . Since ACARS was not functioning properly; we used ARINC and ZZZ3 Radio to patch us to our dispatcher to come up with a plan. ZZZ4 was our immediate alternate to divert to but once we had done our fuel calculations and weather checks; which came to be that we would land with approximately 400 pounds over our reserve fuel and that ZZZ was reporting a ceiling of 3600 feet at ETA; we concluded as a crew and with dispatch that it was safe and best to return to ZZZ. Our main concern was that we had to eventually descend in conditions where there were reports of moderate icing. Through coordination with ATC; their meteorologist who looked for soft spots in icing; and our First Officer and the ice dispersion procedure in the QRH; we came up with a plan of descending through the observed areas lighter in icing at 320 KIAS until we were no longer in icing conditions. Since our A/I system was compromised and the given fact that we had to descend through moderate icing; I had to assume we were going to collect ice on the wings which prompted me to elect to fly at 320 KIAS (slightly under VMO) as directed to do so by the ice dispersion procedure; published in the QRH. We; as a flight crew along with ATC; agreed with the proposed plan. The flight attendant was advised of the situation; and I made an announcement to the passengers that we were diverting back to ZZZ due to the conditions in ZZZ1 and the current state of our wing anti-ice protection. We executed the plan in the descent where we eventually picked up ice and while our Wing anti-ice worked initially; the system then again became compromised despite our N2s being in the required power range. We got down to 4000 feet where we were no longer in icing conditions; since the OAT was 13 degrees though still in IMC and flying at 320 KIAS. Since ZZZ was reporting a temperature of 5 degrees on the ground; I knew to expect a temperature inversion. I requested to fly no slower than 250 KIAS once we were in the terminal area until we were no longer in IMC; which was at around 3000 feet. Once in the clear and 'ICE' not annunciated; we completed the flight normally and required no further assistance from ATC or the ground emergency vehicles and taxied to the gate under our own power. There was an MEL for this aircraft that stated the RH bleed source was unavailable. We ran through the required procedure and ensured the source was from the LH side; before we initially pushed of the gate. I am not sure if this was a contributing factor to our situation; but I believe that it is relevant to include.