Bickell is a third liner. The thing is, calling someone a "third liner" doesn't mean they can't contribute in the top 6 depending on circumstance, especially for short stretches. With how different teams use different lines, one team's second line is another's third.
To put it more bluntly, Bickell is ideally a checking line-type player. He has limited offensive upside but he has enough of it to be able to produce a bit on a scoring line.
If you're adding him to your team based on his playoffs then you're making a mistake, because he's not going to reproduce that rate of production. If you're adding him to your team for what he's shown over his career and are willing to overpay him due to his playoff showing, then you won't be disappointed. Whether that's wise or not is a separate matter, but a young team with plenty of cap space will probably not mind having a solid veteran around the kids.
I'd be willing to pay him $3m in Buffalo. $3.5 is probably too much unless the term is sweet. I doubt he comes to a team like Buffalo without getting at least $3.5m. I'd personally like to see him find a way to stay in Chicago. I'm sick of seeing well-built teams lose out on talent only for that talent to get overpaid and underperform somewhere else.
He needs to stay in Chicago or else he won't exist next season. Take a little less to stay on a solid 4 line team and know your role. Role players excel during playoffs so might as well stay put if possible. If he goes elsewhere for more money, he will try too hard to be something he isn't.