Thank you for your question. Unfortunately there is not an option nor parameter for shortcode to set specific date of check-in for availability form of certain accommodation type. This option is available for global search only.
As a workaround you may place availability shortcode for 6 months of whole year so that your visitors could see when the season actually starts. You may use the following shortcode for this
[mphb_availability_calendar id="777" monthstoshow="2,3"] (you need to replace 777 with ID of your accommodation type)