A TRACON Controller reported a departing aircraft was unable to comply with the required climb gradient and flew below the Minimum Vectoring Altitude and into a potential confliction with traffic departing an adjacent airport. The pilot informed the Tower controller they could not comply but was allowed to depart anyway and the TRACON Controller was not advised.

Narrative

Aircraft X checked on climbing on the ZZZZ departure after departing Runway XXL from ZZZ. I had already released Aircraft Y off Runway XY from ZZZ1; climbing to 020. Because the ZZZZ departure requires a specific climb rate by the aircraft; and the aircraft also must reach [FL] 020 MSL or 4 DME east of ZZZ VOR before making a left turn; it has been determined that a release from ZZZ1 assigned at or below 020 is procedurally separated from ZZZ departures. However; when Aircraft X checked on; I noticed the aircraft was too low (clearly this should have been anticipated anyway; being Aircraft X); and I began issuing a low-altitude alert just prior to the 'LA' alarm going off automatically. I confirmed that the pilot had all terrain in sight; informed him of the minimum altitude; and confirmed he would begin a left turn. The pilot indicated that he was aware he was not achieving the minimum climb rate nor the altitude requirements of the departure. I then called ZZZ1 to attempt to amend the departure heading of Aircraft Y to Heading 360; which was in compliance with the DVA (Diverse Vectoring Altitude) allowance departing Runway XY.ZZZ2 Tower had already transferred communication to me; and I turned Aircraft Y myself once he had checked on. There was not a loss of separation between the aircraft; only because of my heading assigned to Aircraft Y; and because of the chance timing of the situation. However; Aircraft X did enter ZZZ2 and ZZZ1 airspace; because of the low climb rate. ZZZ2 called the ZZZ TRACON CIC (Controller in Charge) to inform of this; then the ZZZ TRACON CIC pointed out Aircraft X to ZZZ1. After the fact; I asked the pilot of Aircraft X if he was aware he would not be able to make the minimum climb rate. He said before departing he informed ZZZ Tower Controller he was unable. He said ZZZ Tower Controller 'gave him the option' of departing south (on the ZZZZZ departure); however the pilot told Tower he would prefer the ZZZ; due to weather conditions to the south. The pilot told me that the Tower Controller had then cleared him for the ZZZ departure. As far as I can tell; there was no coordination by ZZZ1 Tower of this situation; they informed neither ZZZ2 nor ZZZ1. I did not give the brasher warning to Aircraft X; because it was my perception that the pilot had done everything correctly that was required of him (informing Tower he couldn't make the climb; then complying with the Tower's clearance to fly the ZZZZ departure.) He seemed concerned he had done something incorrectly; but I told him he had done it correctly. It is my understanding that ZZZ Tower should never depart an aircraft on the ZZZZ departure when they can not make the climb gradient. Even so; they certainly should not have done so without any coordination or approval from ZZZ TRACON; ZZZ2; ZZZ1; or ZZZ4. In my understanding; coordinating and obtaining approval from all of these facilities would have been required before allowing the aircraft to depart. This created a potentially dangerous situation for Aircraft X; any aircraft in the ZZZ2; ZZZ1; and ZZZ3 patterns; and for Aircraft Y.