Use a Half-Red-Dragon, Half-Black-Dragon Beast Dragon
Or, if you prefer, a Half-Black-Dragon, Half-Red-Dragon Beast Dragon. Or, since there's a way to get Half-Beast-Dragons, like the article says are common, maybe combinations with that, too (though they get two fewer immunities). In any case, you get a creature with all three dragon bits and some powers stemming from each. Bonus points if you make a Half-Red-Dragon, Half-Black-Dragon, Half-Beast-Dragon nine-headed hydra.
This lets you create a creature with some of the abilities of all three, and potentially all the abilities of one of them. Also a truly massive strength score and immunity to, in the titular case, fire, acid, cold, and electricity.
Such a creature can naturally come to exist without outside intervention through normal draconic baby-making, albeit between unusual parents, or perhaps born in an unusual location (a Beast dragon born to a Red dragon and a Black dragon would work, for example), or something else. In any case, the template is certainly clear that it can be applied.
Getting the Titular Dragon as a PC/NPC
All of the Dragons involved, as well as the Half-Dragon template, provide level adjustment should you wish to use them as a player or acquire them via features that are LA-limited. Specifically, a PC may play a level 0 Half-Red-Dragon, Half-Black-Dragon Beast Dragon Wyrmling at 15th ECL.
A PC may also acquire such a creature as a cohort with a leadership score of 21 and the Leadership feat, or as a follower with a leadership score of 71 and Legendary Commander. Older versions can also be gained as a cohort, but not as a follower.
A PC may use True Mind Switch, Dominate Monster, or other magics to gain access to such a creature given the existence of such a creature.
A PC may use Epic Spellcasting and the Summon seed to summon such a creature and press it into indefinite service, even up to the Great Wyrm variety.
An Antipaladin may also follow the rules in the Draconomicon to gain such a creature as a special mount, as could a paladin were such a creature subjected to a helm of opposite alignment.
You can create a duplicate of such a creature by exposing one to a mirror of opposition, though the duplicate ceases to exist as soon as itself or the original is defeated so careful measures must be taken to ensure neither case occurs.
A PC can manufacture such a being in a variety of ways. The first is via Similacrum, which allows you to create partially-real versions of any creature imaginable. While it might seem like you first need a piece of the creature, such a piece should be available in your spell component pouch. The ruby involved, however, costs 100gp per HD the simulacrum will possess, which is half the HD of the creature being mimicked, and you can only mimic a creature whose HD is less than twice your caster level.
A 78 HD Half-Red-Dragon, Half-Black-Dragon Great Wyrm Beast Dragon (the minimum for the mimic to itself function as a Great Wyrm specimen) costs 7,800 gp (and an equal amount of XP) to create and requires caster level 39.
You can also create such a creature with the Conjure, Life, and Fortify epic spell seeds, which allow you to create most any creature.
You can create a mechanical creature, an effigy, via Craft Construct, as detailed in Complete Arcane, but effigy creatures lose all special qualities and most special abilities, and so aren't really very appropriate.
You can use a Sarruhk from serpent kingdoms to grant the various abilities of a red, black, and/or beast dragon onto any Scaled One, like another dragon or a Kobold or a tiny viper. This way lies the path to madness, though.