Let's assume minimum, Moon Druid 4, Pact of the Tome Warlock 3
Wild Shape works by:
You can stay in a beast shape for a number of hours equal to half your druid level (rounded down). You then revert to your normal form unless you expend another use of this feature. You can revert to your normal form earlier by using a bonus action on your turn. You automatically revert if you fall unconscious, drop to 0 hit points, or die.
With level 4 Moon Druid, the PC has 2 hours of wild shape time without expending another use. Moon subclass lets them use only a bonus action to expend a use of wild shape. And a short rest is only an hour. So I am thinking that taking a bonus action every other hour of resting would help here.
Aspect of the Moon is:
You no longer need to sleep and can't be forced to sleep by any means. To gain the benefits of a long rest, you can spend all 8 hours doing light activity, such as reading your Book of Shadows and keeping watch.
Could the druid spend their week of down time in town wild shaped the whole time, and could the same druid stay in a tiny form like a house cat and be carried by the party to stay in wild shape indefinitely until strenuous activity or being dropped to 0 hp occur?