If your systems best is DME/DME, don't forget that at beginning of T/O roll, often your equipment will make a runway position update - the accuracy that follows from that update normally satisfies SID requirements.
However, if your system is not CERTIFIED for RNP-1 (FAA AIM Chap 1, Sec 2, par 1-2-2), you cannot LEGALLY fly an RNP-1 procedure, despite the fact that the equipment may [temporarily] have the required accuracy. On-board monitoring capability is an integral part of RNP certification. Also, FAA Advisory Circular 90-100A is very explicit that aircraft certified under earlier AC 90-45A RNAV criteria (prior to 2005) may not be compliant with the standards required for current RNP-1 procedures.
the local authority had approved the airplanes as RNP5. Now, since RNAV departures require P-RNAV or RNP1, I was wondering if we are authorised for RNAV departures.
You cannot legally fly an RNP-1 RNAV departure with only RNP-5 certification. There may be airports outside the US and Europe that do not require better than RNP-5 for SIDs, but I don't know them off hand.