I'm specifically suggesting transforming the integer in an intelligent way to map well to the mantissa of the floating point representation, based on the intend

I'm specifically suggesting transforming the integer in an intelligent way to map well to the mantissa of the floating point representation, based on the intended usage. High frequency animations can chop off the most significant bits, while low frequency animations can chop off the bottom bits. If you aren't able to constrain the design or make assumptions about frequencies, then it seems like you have to instead parameterize time as some sort of tuple so that you have more bits, such as the sum of a course absolute and a fine offset, and write the shader to cope with it. That's how time APIs in many operating systems work, where there's integer seconds and integer micro or nanoseconds.