Rayse has killed 4 vessels at this point, and does not seem to have overt support from other Shards in doing so. I was thinking about a wob where Brandon was asked about if Autonomy assisted Odium in his attack on Dominion & Devotion and says
The idea that Autonomy's assistance was what allowed Odium to win a 2 v 1 contest has never seemed satisfying to me, as even a 2 v 2 does not seem like a certain thing... When combined with the fact that Hoid asks an Avatar of Autonomy for help with Odium (which would seem silly if they'd overtly worked together in the past) it seems that Odium would needed some other (stronger) advantage to ensure victory. A Dawnshard seems to fit that description. I've also wondered why Honor implies things are (nearly?) hopeless in the fight against Odium without the Dawnshards even though Cultivation is still around. However, if Odium has/is/holds a Dawnshard, then it seems more reasonable that the only way that Honor could 'win' would be by tricking/trapping Odium in the Rosharan system, as a direct fight against a Shard+Dawnshard was not something he felt even Honor and Cultivation together could take on.
As there is not a large amount of evidence for this theory (some would say none), I'll try to argue against it: If there are only 4 Dawnshards (it seems there could be 16?) then there are 2 (or 3 if Dalinar is/holds a Dawnshard) of the 4 in the Rosharan system, which seems like a lot. We also don't see anything that jumps out at me in the letters to/from Hoid that suggests to me that Odium is MORE powerful than other Shards. Everyone wants him contained because of what he has done, but not specifically because he's more powerful than anyone else. Additionally, Brandon has said that Rayse won't pick up another Shard because he doesn't want to change and Brandon also says that Dawnshards change your spirit (though who knows how this would work with a Vessel).
All in all, I'd say that Rayse having/holding/being a Dawnshard makes more sense than him not, but we know so little about Odium's battles with other Shards that the questions posed could be answered in any number of ways. I'm interested to hear some thoughts if you have them, or if anyone has any of that pesky evidence it's so hard to come by (for or against of course) I'd love to hear about it!