Are there DC removal techniques that do not introduce a filtering transient?
Any filter with a DC gain of zero and, for most other frequencies, a gain of (or close to) 1, is a DC blocking filter.

So just make a high-pass filter (HPF) and put the corner frequency as close as you can to zero.

Of course, when the system is switched on, there will be a transient from blocking DC of 0 to blocking DC of whatever value it is.  Without prior knowledge of what the initial DC is, I do not see how you can assume the initial value is anything other than zero.  So all of the initial states of your HPF are zero.  But once the turn-on transient is over, there should be no other transients unless the DC component has a sudden change (like a step function) and is not just "trending" to another value.

If this is done in fixed-point, you will have some serious numerical issues.  You might even have serious numerical issues if it's floating-point and your HPF corner frequency is very very low. You might want to look at this old thing from me or this article from IEEE Signal Processing magazine.  I dunno where to find it free.