In a short answer:
There is no need to calculate the days to such precision, and it's just not efficient. There's some complex math and technology needed for that precision that wouldn't exist in your renaissance-like time period.
What time measurements am I using?
To begin, realize that if the your answer to this question involves seconds, minutes, etc., these terms need to be defined.
On Earth, a "second" is defined as 9.19 billion oscillations of the outermost electron between two electron energy levels of a cesium-133 atom. Let's call this measurement an "EO".
Would your renaissance-esque civilization be able to calculate an EO?
Let's take a look at your calculated sidereal year. You defined a sidereal year as:
435.126 days long
Again, the term "day" is ambiguous. On Earth, that would be 86400 seconds (7.94 trillion EO).
Now assume that you are using the EO in your world. To precisely calculate to 0.126 days, you need to be able to calculate to milliseconds; one thousandth of a second.
Who sets the standards for time?
In your world, who/what is keeping track of the time? Is there a global standard of time like we have here on Earth? Who/what is dedicating its time to sustaining this time-keeping?
Note that here on Earth, our global standard is a manufactured product that has to be tweaked every so often. This results in our TAI standard (that is used for scientific purposes) being different than our UTC standard.
As of 30 June 2015 when the last leap second was added, TAI is exactly 36 seconds ahead of UTC. The 36 seconds results from the initial difference of 10 seconds at the start of 1972, plus 26 leap seconds in UTC since 1972.
There's also the concept of time zones.
A good example would be in the 18th and early 19th century. If you were to travel, you would have to reset your watch at every stop along the way; a tedious task. The creation of time zones meant that a large area would all be on the same time, so if you traveled from San Francisco to Chicago, you would only need to change your watch twice.
Where does time begin?
There's a lot to consider here.
- When does your timing system begin? Where is your 00:00:00?
- Where do days begin/end? months (if you use them)? years?
- How are the years being counted? Starting from zero? one? a negative?
- Would you use a likeness of a BC/AD system? If so, what events define where one ends and the other begins?
Oh, and here's a fun one.
What about time dilation?
In the theory of relativity, time dilation is a difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from a gravitational mass or masses.
Not only is this a incredibly tough problem, it brings to mind the issue of miscommunication between people in your world.
Unlike today, a renaissance-like time period wouldn't be able to instantly share data like we can today with the Internet. It could take days, months, or years for them to communicate across the globe; leading to unavoidable inaccuracies in their data.
In conclusion, you either need to create your own time system (and maybe universe), or figure out a reasonable way for your world to calculate time to the millisecond. Don't forget that there's a whole lot of thought that needs to be put into creating a new time system.
I hope you found what you're looking for. Good luck!
The only reason for time is so that everything doesn't happen at once. -Albert Einstein