Two things: those 'cooler heads' are also people whose policies we don't like. Example: Cheney - oil, military pork, supporting whatever nasty regimes his bosses find useful.
Second, Reagan didn't seem to suffer from opposing labor. Why should Bush? Especially as 90% of the assault can be done in indirect ways, such as judicial appointments, clauses buried deep in 1,000 page congressional bills, NLRB appointments, Dept. of Justice prosecutions, etc.
Barry