is the 1/4" bevel supposed to be the leg or hypotenuse distance, I wondered.
The 1/4" is the leg size, but this is perhaps a bit more complicated answer than you realize (assuming AWS D1.1 is the design standard).
The welding symbol that you are showing is somewhat incomplete (putting the 1/4" dimension in parentheses would fix that). Typically as an engineer you are concerned about the strength of the weld, and not as much about how that is achieved. So the important issue would be to callout that you want a PJP groove weld with an effective throat of 1/4", this would be done by putting the dimension in parentheses. The fabricator would then pick their weld process (SMAW, GMAW, etc.), the bevel angle and bevel size to achieve required effective throat. For instance, for a prequalified PJP weld and a 45° bevel, if GMAW is used in the flat or horizontal position, the effective throat is equal to the bevel size (1/4" bevel gives 1/4" effective throat), but if SMAW it is 1/8" less than the bevel size (1/4" bevel gives 1/8" effective throat).
Without the parentheses around your 1/4" dimension, you are calling out a 1/4" bevel but you aren't dictating the weld process, so it isn't clear what effective throat is actually needed. I would guess the fabricator (conservatively) assumed your dimension was meant to be the effective throat size and they are using the SMAW process so to achieve the 1/4" effective throat they need a 3/8" bevel in order to be prequalified.
Be aware that AWS D1.1 does not consider a weld made without backing or back gouged to be prequalified. To make a weld without backing or back gouging, it is considered partial joint penetration. The contractor would have to qualify the WPS and the welders without backing in order to comply with AWS D1.1.
Perhaps it's the wording, but this doesn't ring true to me. PJP welds are absolutely prequalified and don't require backing or back gouging. I'm assuming you were talking specifically about CJP welds here? As the HSS sidewall is 1/2" thick and the OP only called for a 1/4" bevel, I think it's safe to assume they are going for a PJP.