Good question - While MBC2015 doesn't specify that the start of multiple interval doublings must be started at the same point, I tend to air on the side of clarity and not double a new interval while in the midst of another doubled interval.
The Introduction to Music Braille Transcription, 2nd edition by DeGarmo, advises that "it is better not to double more than one interval, unless the doubling begins and ends simultaneously for all the intervals..." (pg 360)
So while there is no rule that says you can't, remember that even the procedure of doubling is not required - we MAY chose to use doubling if it remains clear and is truly helpful. Once you start doubling multiple intervals, the complexity could very quickly outweigh the benefit of the procedure. It's not always about just saving space - it's about making the music easy to memorize and perform/study.
Hope that helps!