PaulS:
Only for debugging purposes. The real intent of the program is to read serial data from Serial3 and send it to a client. That uses SPI. So, the question of the speed of SPI vs. the speed of Serial is a quite legitimate one.Of course, as has been seen, the issue of SPI speed is not the limiting factor. It is the overhead of packeting one character at a time that is/was the problem.
Granted that it's only for debugging purposes, but it is having the effect of throttling the whole sketch to the speed of the output stream, which will result in the input stream overflowing.
I don't see any reason to suppose that the overhead of packeting one character at a time over SPI is causing any problem. The Ethernet interface, and the SPI bus used to access it, and the processor which is performing the work, are all capable of dealing with far greater throughput than the serial stream.