OTOH, finding caps with a high enough ripple current rating that aren't low ESR, and/or 105 deg C rated can be difficult. I created a quick & dirty sim of the O.P's cap at the specified 740mA load current, and it looks like it will need a minimum ripple current rating of 1.3A RMS. That's easy to find in a low ESR 105 deg C radial format, but much harder and more expensive to find in axial format with an ordinary ESR. The sim gave the peak surge current as about 18.5A for the first half cycle at 10% high mains voltage, and 50% excess capacitance, assuming the only limiting factor was the transformer secondary resistance (all other components ideal).
The O.P's schematic doesn't identify the rectifier diodes, so lets assume they are generic 1A silicon similar to a 1N400x series diode. A 1N400x diode's peak non-repetitive surge rating is 30A for a single cycle, and by the next cycle the peak's dropped to 6A worst case and its within its 1A average continuous rating. Therefore I'd have no concerns with replacing that cap with a low ESR one unless the diodes looked really wimpy or already showed signs of overheating, which would encourage me to replace them as well.
If its for your own use, bodging in a radial cap in place of an axial is fairly simple if you crop whichever lead will go to ground and splice on a length of solid core insulated wire, bent over the rim and secured with a small cable tie near the top of the can. If you bend the lead over, you *MUST* support it at the bend so you don't stress its seal in the bung as any damage to the seal typically results in premature failure. For liability reasons don't bodge it with a radial if you are being paid for the repair.