So, from the text as written, it sounds like if you have specialization in both punching and the cestus, you get to apply the bonuses for only one of them each round, but you can choose which one round to round. It's honestly not very well written, but to me, it reads that the tradeoff is that punching specialization only gives +1 to damage instead of the normal +2, but instead you also get a +1 to the result of the "Punching and Wrestling Results" table (page 129 in the PHB).
Which is a pretty dubious bonus, to tell the truth. The only thing you're getting from the chart with the cestus is the chance to KO, but that chance doesn't have any relation to how well you roll. In fact, it's practically random, with the best possible chance (25%) requiring your modified attack roll to be 1 or lower.
Essentially, from my read, what you get with the cestus is a weapon that is easy to use (doesn't require proficiency, 1 point to specialize), can be versatile if invested in, and allows (clumsy) grappling without having to drop your weapon. Personally, I wouldn't give it the tumbling bonus to unarmed attack rolls for the same reason that the description gives it a negative to grabbing and holding: the character isn't truly unarmed. Also, given how specific the description is otherwise, I feel that they would have mentioned the tumbling proficiency if it applied.