ARGYLE, Texas — His famous name shadows George P. Bush, the only member of the dynastic political clan now in public office, as he enters the final days of an uphill campaign to unseat Texas’ attorney general. To some Texans, the Bush family name is a badge of integrity, harking back to a bygone era of rectitude and respectful political debate. To others, it is the disqualifying mark of a Republican old guard that failed the party and betrayed its last president, Donald J. Trump. Mr. Bush would like to make the campaign about the two-term Republican incumbent, Ken Paxton,...