A game mode like this really forces participants to use ancient/normal spellbooks for magic, specifically. Keep in mind that this would make it heavily hybrid/tribrid-dependent. Based on your initial idea, dying once would then disqualify you from the tournament, correct? This would create a large disinterest in participating from those preferring melee over hybrid.
If you want a game mode like this to be fun for all of your players, I would recommend the following tweaks:
Enable the ability to respawn (in a random location) and continue fighting. This would make it a true FFA mode, where you can keep fighting as many opponents as possible under a specified time limit. There would be an actual competition of players fighting for the most kills, rather than who's able to stay alive the longest.
With the previous in mind, the gear you choose to start with would be your permanent gear throughout the duration of the tournament (meaning you would respawn with the same setup & inventory after each death). You can, however, loot gear from others players and use it if necessary.
Disable the use of any PKP items. This would give everyone an equal opportunity at winning. Newplayer473 shouldn't feel discouraged because they don't have decent gear.
Disable overhead protection prayers. These honestly just prolong fights, meaning less time for you to get kills. If NHer398 is busy fighting someone for 5 minutes with constant overheads and prayer switches, he's wasting time while Propker1337 is killing people in 90 second fights.
Require at least two out of the three combat types to participate. This meaning you must have some combination of melee/ranged, mage/melee, or mage/ranged weaponry (something as simple as having a rune crossbow w/bolts in addition to their magic/melee weapon). This would shutdown anyone trying to farcast as protection prayers are disabled and their opponent is guaranteed to have either magic or ranged.
These are suggestions from a non-biased perspective.