Try loosening your laces a mile or two before your feet would normally start to hurt. Feet often swell, and if you can loosen the shoes after they start to swell but before the pressure begins to cause irritation, you may be able to head it off.
Loosening at the very start of a ride might annoy you and drive you crazy as your feet would slip around in the shoes at the start, but you could also try that.
Clipless won't change your physiology. If you're someone whose feet get nice and juicy when you ride, they will still get nice and juicy regardless of the pedal type.
You could always ask your podiatrist for shoe recommendations for the bike. I'm quite fond of riding in sandals due to my feet's personalities. (Chacos and Keens) She would know where she's seeing tendon issues in you and what kinds of shoes might help minimize the effects.
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