Flux-tunable qubits are a useful resource for superconducting quantum processors. They can be used to perform CPHASE gates, facilitate fast reset protocols, avoid qubit-frequency collisions in large processors, and enable certain fast readout schemes. However, flux-tunable qubits suffer from a trade-off between their tunability range and sensitivity to flux noise. Optimizing this trade-off is particularly relevant for enabling fast, high-fidelity, all-microwave cross-resonance gates in large, high-coherence processors. This is mainly because cross-resonance gates set stringent conditions on the frequency landscape of neighboring qubits, which are difficult to satisfy with nontunable transmons due to their relatively large fabrication imprecision. To solve this problem, we realize a coherent, flux-tunable, transmonlike qubit, which exhibits a frequency tunability range as small as 43 MHz, and whose frequency, anharmonicity and tunability range are set by a few experimentally achievable design parameters. Such a weakly tunable qubit may be used to avoid frequency collisions in a large lattice while exhibiting minimal susceptibility to flux noise.