Is it possible to have the start of the rental to be only within 4 days from the moment of booking, but without restrictions for the end of the rental?

Yes. Copy your existing rule of work. In the rule created, unset the Show in colors section checkbox. And set the rule parameters so that it Ends (from the moment of booking) after 4 days.
After that, make sure that a new rule has a higher priority (it should be located on the right side).
Then create a new tip of Error type. Specify the message saying that the booking is possible only withing 4 next days. Unset the checkbox Apply even if there are missing rental intervals and disable this tip by default.
Enable this tip only at the old rule of work.
As a result, when customer books the time at the intersection of the rules, the tip will not be displayed. And when he is booking only on the old rule (e.g. from the 5th to the 10th day from the current moment), an error will be displayed.
See an example in action:
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