Just to add, it turns out the filesystem isn't in low memory mode, so I'm afraid it can't go any faster. So I guess the only options are:
Increase the input buffer size
Re-write the SD library so that it's nonblocking - however that would require the FAT library to be rewritten too!
Speed up software SPI (it could be implemented like jswrap_io_shiftOut, which would really speed it up by accessing the register directly - at least on STM32). That may still not help you though
Don't use an SD card. For instance you could use SPI flash, and then you can create your own journalling filesystem that wouldn't have these pauses.
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Just to add, it turns out the filesystem isn't in low memory mode, so I'm afraid it can't go any faster. So I guess the only options are: