A 14-4 defeat in tonight's elimination play-off ended the Dragons' season and signalled the end of the road for popular Australian second rower Menzies, who hung up his boots and trademark headguard after 21 years as a professional.
"It's an emotional dressing room," Frayssinous said. "It's disappointing that we say goodbye in that way.
"He has been one of the best overseas players we've had at the Dragons, if not the best, and we wanted to give him at least one more week."
Menzies had a late try disallowed to compound his misery as the Dragons sorely missed the creativity and kicking game of scrum-half Scott Dureau, who pulled out with a quad muscle injury.
"We missed the vision of Scott and his kicking game as well," Frayssinous said.
"It's very disappointing that we couldn't execute, especially with all the possession we had. Hull FC defended pretty well, so credit to them, that's why they won."
Hull coach Peter Gentle paid his own tribute to Menzies, who is to return to his Australian club Manly Sea Eagles to take on an ambassadorial role.
"He's held in high regard throughout the world," Gentle said. "I've nothing but admiration for him.
"He showed tonight he can still play at this level. He could play for another couple of years."
Gentle claimed his side put the lessons from their recent Challenge Cup final defeat to good use as they stayed alive in the play-offs.
The Black and Whites failed miserably to cope with a greasy ball at Wembley, where they lost 16-0 to Wigan, but adapted to the wet conditions at the KC Stadium by defending superbly.
"We learned a lesson about a month ago about playing in these conditions," Gentle said.
"I can't doubt the courage and character of the group. There were players entitled not to be out there tonight. We are down on troops, there was a lot of courageous efforts out there.
"It was a great team performance but I thought our senior players were terrific."
Full-back Jamie Shaul, 21, provided the highlight with an 80-metre opportunist try, collecting Leon Pryce's towering kick on his own 20-metre line and sprinting through a fragmented defence for his sixth touchdown in just seven Super League appearances.
"He can do that," Gentle said. "He is blessed with speed. He's learning his trade. He is nowhere near the finished product but I'm very happy with what he's doing.
"He's been sitting on the sidelines for most of the year and had to wait for an injury to get a start but he didn't drop his chin."