Is it possible to have a high resolution FFT.

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If I understand you correctly, that is not going to work. I haven't used an Si5351 but I don't see how you use a frequency from the Teensy to "tune" the Si5351. The Si5351 is a clock generator, not a frequency synthesizer.
You would be a lot better off implementing the encoding of the WSPR digital signal directly in the Teensy and then use that to drive a synthesizer chip.

I have generated JT65 signals with a Teensy++2 and a DDS-60 which uses an AD9851 synthesizer to transmit on 40 metres. I've also implemented FT4 audio encoding in a Teensy. Neither of those is a trivial thing to do but they are a whole lot easier than trying to make the Teensy discriminate between four tones which are separated by 1.4648 Hz and then somehow use that to drive a synthesizer.

Pete
 
I am working odf experience using the SI5351 for standalone WSPR. I would be using the Teensy and the Si5351 arduino library to set the frequency. https://github.com/etherkit/JTEncode/blob/master/examples/Si5351JTDemo/Si5351JTDemo.ino shows an example of someone's program for directly sending from the arduino using the SI5351.

I could just skip all this and wait until I have enough money to actually afford a HF radio (or build one) but this seemed like a fun project to try and work on.
 
I stand corrected. I hadn't expected that I2C would allow sufficient control to maintain the symbol timing.
You would certainly be better off using the encoder to generate the WSPR signal.

Pete
 
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