The Bears defeated Toulon 32-19 in Aix-en-Provence to become the ninth English Premiership team to take the title.
Harry Randall and Max Malins scored attempts and Callum Sheedy, half of the fly, landed 22 points with a perfect eight of eight successful kicks on goal.
The win came a week after Bristol lost to Wasps in the Premiership semi-finals, but Lam confirmed they were put on standby to play Exeter Chiefs at the Twickenham show after Wasps had seven positive coronavirus tests.
“We obviously want Wasps and Exeter to meet in the final because they really deserve to be there, and that’s the final that should happen,” said Lam.
“But in the world in which we live with COVID, we don’t know. We have been told that we are ready. So if we’re told we’re in the finals, we’ll bring everyone together and say let’s go.
“It’s just the world we live in right now. Things are always changing, but we definitely hope Wasps and Exeter make it. ”
Exeter wants to follow Bristol’s example
The Chiefs hope to achieve an English doubles in Europe by beating Racing 92 for the first time in the Champions Cup on Saturday at Ashton Gate.
Now back home, Bristol will no doubt keep an eye out for what’s happening instead of celebrating a successful end to their season.
“I said to the players before the game, let’s go out and celebrate who we are and what that means. The way we play the game, our culture, the love we have for each other, ”said Lam.
“The focus was on a little joy. We knew we would be under pressure against a massive team from Toulon. We did it by sticking to the Bear Way.
“The players did it on behalf of the entire squad and the saddest thing was that we couldn’t have everyone with us – Steve Lansdown and our captain Steve Luatua. Congratulations to him and his wife Olivia on having a baby girl 45 minutes before kick-off.
“Our whole vision was to inspire our community with rugby successes. This is a moment of inspiration, but we have to go back and get better.
“We want to get better and grow with every game, but there is still a long way to go. It is still good for our development to have won a really good trophy. ”
England flanker Ben Earl, on loan from Saracens, was named Man of the Match while Sheedy Wales manager Wayne Pivac kicked off a week before Wales clash with France in Paris.
“It is one of the first trophies Bristol won. I do not know how long. We were so calm and controlled out there, even when we lagged behind, and we trusted our system, ”Earl said.
“I’m just so proud of the guys and our defense showed the character of the team. We have had a difficult journey since we were locked down together – some highs and then huge lows like last weekend.
“The mood in the camp after the result of the wasps was very bad and it took a lot to get back on your feet. But the group was confident that we would get the job done.
“You could say that the feeling was there from the start.”
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