Excellent question, i hope to learn more from incoming answers.
One thing i found is that although the lunar months are easy for anyone to follow - regardless of whether they have a calendar or not, the Moghul government would use the lunar calendar, but utilise the solar calendar for taxation due to it's congruence with the seasons and harvests.
The seasons and harvests do appear to show that the solar calendar is not something to be discarded completely in calculations or estimations (although the solar calendar gets adjusted in bulk once in a while to compensate for the fact that the solar year doesn't end at exactly 365 days (i read it's 365.2422 - so even the leap year adjustment can't do a classic timepiece job) - and as far as i can see - the adjustment of official months (possibly in context of leaders commiting corruption) is not allowed.
إِنَّمَا ٱلنَّسِىٓءُ زِيَادَةٌ فِى ٱلْكُفْرِۖ يُضَلُّ بِهِ ٱلَّذِينَ كَفَرُوا۟ يُحِلُّونَهُۥ عَامًا وَيُحَرِّمُونَهُۥ عَامًا لِّيُوَاطِـُٔوا۟ عِدَّةَ مَا حَرَّمَ ٱللَّهُ فَيُحِلُّوا۟ مَا حَرَّمَ ٱللَّهُۚ زُيِّنَ لَهُمْ سُوٓءُ أَعْمَٰلِهِمْۗ وَٱللَّهُ لَا يَهْدِى ٱلْقَوْمَ ٱلْكَٰفِرِينَ
Verily the transposing (of a prohibited month) is an addition to Unbelief: the Unbelievers are led to wrong thereby: for they make it lawful one year, and forbidden another year, in order to adjust the number of months forbidden by Allah and make such forbidden ones lawful. The evil of their course seems pleasing to them. But Allah guideth not those who reject Faith.
The principal astronomical cycles are the day (based on the rotation of the Earth on its axis), the year (based on the revolution of the Earth around the Sun), and the month (based on the revolution of the Moon around the Earth). The complexity of calendars arises because the (solar) year does not comprise an integral number of days or an integral number of lunar months
The civil calendar in use around the world (Gregorian calendar) is a solar calendar. Solar calendars are based on the progression through the seasons as the Earth revolves around the Sun, but neglect any attempt to keep the months synchronous with the lunar phases. A lunar calendar bases each month on a full cycle of the Moon's phases (called a lunation or synodic month) without regard to the solar year. Lunar calendars usually start each month with a New Moon or the first visible crescent moon after New Moon. Luni-solar calendars try to remain synchronous
with both the solar year and the moon phases. However, a solar year does not contain an integral number of days or an integral number of lunar months. To compensate for this, many luni-solar calendars adjust the length of their years and months. Without such an adjustment the seasons will steadily drift through the months.
Another question that arises in my mind is:
How on earth (or in heaven) can it all be expanding, and slowing down, yet be 365.2422 (24 hour days) all at the same time?