The key in the advice in the data-sheet are the words "at least".
You will have to keep an eye on thermal management if your space is very limited. But other than that, the 15.4V of the rule of thumb is the minimum advised voltage, where 20V, 25V, even 500V would be fine.
Another rule of thumb is, that a very low Vf (forward voltage) is easier to find at the same forward current with a lower reverse breakdown voltage. So you'd be easier off finding a suitable device with Vbr = 30V than at Vbr = 300V.
I'd say a diode with Vf =< 0.4V at 5A should be possible to find at Vbr = 25V, you may have to look at diodes that can handle up to 15A or 20A, though. (Again, 5A is the minimum specified, higher never hurts).
Diodes are complex things, so I have taken the liberty to suggest one by having Farnell sort by price within given specifications (as your example was through hole, I'm skipping SMDs):
Has a nice data-sheet showing in a graph that it approximately has a Vf of 0.42V at If of 5A. Still, that's quite a lot of power to dissipate, if the 5A is a regular thing, as stated, you should find a way to cool it. One option is lots of copper space connected to the pins.
As an example of my point, from the same series, the device that handles 15A in stead of 12A, the Vf goes from 0.42V to 0.4V at 5A (see data-sheet, graph on page 2, left side, second from the top):
And if you could fit one in, this TO220 type makes cooling a "breeze", by just adding a heat-sink, if that's necessary, as it again has a slightly lower Vf. I have to estimate it to be about 0.30 to 0.33V at 5A, because the data-sheet has no fancy graphs:
I hope this clarifies some things for you.
My exercise of increasing cost for lower Vf/higher Current can be done with many different supplier websites, such as Mouser, DigiKey (though I find their site more annoying), or as I did, Farnell. Keep in mind, you want low Vf and the Vf rated on the cover sheet / supplier page is almost always at its rated current, where a lower current means a lower Vf. So a device of Vf = 0.5V at 25A rated will become slightly less hot at 5A than one rated for Vf = 0.5V at If = 5A.