The usual way to separate conjugates from non-conjugates is using HPLC, but that's a very specific and expensive setup. Your condition or only using size implies use of size exclusion chromatography. Now your peptide has molar mass of about 2.5kDa, while oligonucleotide is between 6 and 30kDa. You will need to pick a column with specific polymer, the most common being Sephadex G-series (consult this article for more detail https://pmc.ncbi.nlm.nih.gov/articles/PMC7121854/ ). Removing leftover peptide with it is straightforward, but for separating DNA and DNA conjugate you will need some sort of salt buffer to prevent secondary structure formation, possibly a solvent mixture gradient and a detector that can track 260nm and 280nm waves as conjugate will have increased absorption there. Sorry, can't give more detailed advice at the moment.

If you're able to play with your AA sequence of your peptide a bit you could have a terminal Cysteine after a few alanines.

This would allow you to immobilise say on a resin or membrane the the AA with a cysteine cleavable linker.

Carry out your conjugation with the DNA and wash the membrane/resin.

This would leave you with 2 species to separate, the conjugated and unconjugated peptide, this would be far easier than just the 3 species (DNA , AA and conjugated mateiral).

You would then just need to find a resin that has an exclusion limit useful for the group separation of the conjugated material, the aa would simply wash through.