I find it interesting that you looked at the question as one about performance qualification and I looked at it from a procedure qualification standpoint.
As for performance qualification, I believe the engineer would still have a problem because the base metal (AISI 4130) isn't included in the list of prequalified base metals.
The problem might have been avoided had the 4130 been purchased to an ASTM standard, but still there is the issue of the base metal being a listed base metal in D1.1 for either procedure or welder qualification.
I would think that in the grand scheme of things that if a welder can pass a test with the higher alloy, higher strength steel, he will be able to pass a test with a low strength steel. For the most part, AWS D1.1 follows that philosophy, but as always, the devil is in the details.
An alternative may have also been to refer to AWS B2.1 for the welder qualification. The material would fall into the M4 (P4) group (1 chrome 0.5 moly) while the A514 Gr B is a M11B (B2.1). Like ASME, AWS B2.1 would allow the welder to be qualified using a wide range of base metals meeting M1 through M11 groupings. Likewise, the F number of the filler metal could be compared. It is most likely going to fall into either a F4 or F6 groups. The welder would then be qualified for any of the filler metals in the F group. AWS D1.1 does have provisions for welders qualified in accordance with B2.1 to be accepted by AWS D1.1 (2004 is the one I looked at). There is a high probability that you can cross reference the 4130 with a UNS number that is listed in AWS B2.1 and work around the issue of welder qualification that way.
The A number isn't a consideration for the welder qualification, whereas it is for the procedure qualification. In this case, the A number of the filler metal may be different for the two base metals listed (assuming the procedures would not use the same filler metal for both base metals), so again the procedure would fall through the cracks.
Details, details, details. Are we dealing with procedure qualification or performance qualification? You may have hit the mark Lawrence and you've given me a reason to revisit my initial response.
Best regards - Al