I think you might be mixed up. Oats have carbs, and little protein. But yes, they would be a good addition. You said when you skip exercise, you eat nothing, is this a regular habit, or just random? If regular thing, then eat your 5000 cals on non training days also. You can do two things, scale back the amount of cardio you do, just out eat the problem. I would choose the second option, since you will need endurance for rugby.
To get to 100kg, that's a fairly large amount to gain. I don't know how much a huge amount of cardio you actually do, but maybe cut it back to half to start. Also, what is your typical diet plan, be specific.