I don't know, there must have been better options than killing Sho and handing the Control Metal to Chronos.
Case in point, they're both able to infiltrate Relic's Point to rescue Sho's dad. Agito could've used a similar strategy to get Sho inside Chronos Japan's HQ.
Yes, but again: he was on a clock. Chronos was moving, and moving fast, and he didn't know his capabilities nearly as well as he does now. I mean, this was, what, days? Weeks after he became a Guyver? He couldn't have spent much time bioboosted. Transforming is a pretty explosive process, and he must have assumed that Chronos was keeping a look-out in the area for just that. Plus he was the son of the area manager! He couldn't very well sneak out, bio-boost and work out for a bit in the back yard.
And would he have gotten Sho to trust him? Without revealing who he really was? "We must sneak into Chronos Headquarters together, Guyver One." Sho might have gone for it, but maybe not. Better not to give him the option.
Say that it would have failed, and Chronos had obtained the control medal. They still did not possess bio-booster organism samples, unless they got some from Guyver-2 (actually I guess they must have, otherwise they could not have created Guyver 2-F's unit - but that success was years away), and Agito did not know about Guyot's plan either. In any event, it would have bought him time. In a pinch, he could have attempted to destroy Chronos Japan from within in under an hour. After all, he was surprised that Guyot was able to resist him in human form... and there were no Zoanoids besides Enzyme who could defeat him.
As said, Guyver-1 was expendable... indeed, a possible threat, as he did turn out to be. After Chronos Japan was destroyed, he went out of his way to get Sho as an ally. But not before. So he was really very much playing chess.
He must have felt really shocked when the pawn upgraded so fast.