Superconducting quantum technologies require qubit systems whose properties meet several often conflicting requirements, such as long coherence times and high anharmonicity. Here, we provide an engineering framework based on a generalized superconducting qubit model in the flux regime, which abstracts multiple circuit design parameters and thereby supports design optimization across multiple qubit properties. We experimentally investigate a special parameter regime which has both high anharmonicity ($sim!1$GHz) and long quantum coherence times ($T_1!=!40!-!80,mathrm{mu s}$ and $T_mathrm{2Echo}!=!2T_1$).