Clint made the right call, amazingly enough. Bucky wasn't just being avoidant of the issue when he said he didn't know what to say. He just... didn't know what to say. It'd been a few days but it was still a very raw, sore state of emotion that he couldn't put to words yet; he didn't want to put to words because he didn't think there were any. He'd been through an internal journey through hell trying to sort those emotions out for the past few days and he wasn't getting anywhere.
The mention of a dick cake made his lips twitch, but that was about as far as his expression got before it died and reverted back to its state of stoicism. If he let himself feel anything at all, it was going to bring the pain back again and he just needed a break.
He didn't tense up or pull away when Natasha took his hands. His legs were a bit stiff but he did stand up. Much as he was pretty sure he could've sulked forever, Bucky could appreciate what the two of them were trying to do and fuck, he wanted to cling to any thought that maybe he wasn't actually alone. At least for the next little bit; long enough to take a shower and eat some dick cake. He couldn't promise he'd say much, or do much, but he did stand up and that was a start.
Slowly, he turned back toward the entrance way into the house. One of his hands fell out of Natasha's hold but the other stayed in her grip as he walked in, which led into the kitchen. Wordlessly, he opened a drawer where there were a few forks and utensils. Bucky stood there and looked into the drawer, for no particular reason other than that he was just trying to focus; keep moving. Now that he was up, he needed to keep moving or else he'd slump over again and he might not have the strength to keep control over the feelings that would accompany it.
He slowly let go of Natasha's hand and took a few seconds just to breathe. He turned his head again, to look at the both. "You can sit, if you want." Bucky didn't have a lot in the way of furniture but he'd built some chairs and a table. His eyes traveled to Clint's face, where he finally saw the bruising and concern etched into his eyes briefly, even though he was clearly okay.
That said, he left the room to go shower. It was more like he was now on autopilot, going through the motions because he had to. They didn't need to see how devastated he was, or how in the beginning he'd been too depressed to even move. Clint and Natasha had plenty else going on; they didn't need this too. They didn't need him bringing them down.