Michael Kruger, a Reformed biblical scholar, has a post arguing in favour of the proposition that the Epistle to Diogentus reflects forensic soteriology (link). It is amazing to see such texts isogeted from their fuller context, though this is reflective of how many apologists abuse patristic literature (cf. William Webster et al. quote mining Athanasius to “prove” he held to sola scriptura, though he held to beliefs no proponent of sola/tota scriptua would accept).
A Catholic apologist, using the moniker “Matt1618” wrote a good refutation of the claim that the early Church Fathers taught Sola Fide; here is what he wrote on this specific text:
For what, save His righteousness, could cover our sins? In whom...could we be justified, save in the Son of God ALONE?"
The author of the Epistle to Diognetus 8 - He sought to form a mind conscious of righteousness, so that being convinced in that time of our unworthiness of attaining life through our own works, it should now, through the kindness of Gob, be vouchsafed to us; and having made it manifest that in ourselves we were unable to enter into the kingdom of God, we might through the power of God BE MADE ABLE. (ANF, vol. 1, p. 28)
The author of the Epistle to Diognetus 8 Having therefore convinced us in the FORMER TIME THAT OUR NATURE WAS UNABLE to attain to life, and having now revealed the savior who is able to save even those things which it was formerly impossible to save, by both these facts he desired to lead us to trust in his kindness, to esteem him our nourisher, Father, Teacher, counselor, Healer, our Wisdom , Light, Honour, Glory, Power, and Life, so that we should not be anxious concerning clothing and food.
He who takes upon himself the burden of his neighbour; he who, in whatsoever respect he may be superior, is ready to benefit another who is deficient; he who whatsoever thing he has received from God, by distributing these to the needy, becomes a god to those who receive his benefits: he is an imitator of God. Then thou shalt see, while still on earth that God in the heavens rules over the universe; then thou shalt see, while still on earth that God in the heavens rules over the universe; then thou shalt begin to speak the mysteries of God; then shalt thou both love and admire those that suffer punishment because they will not deny God (ANF, vol. 1, p. 177)
Conclusion - In this very short letter, there is not much in regards to justification. But of what is there, we see that what justifies is a making just, not a mere declaring just. Our own righteousness and works without grace merits nothing before God (Trent, canon 1, justification). God's grace does not merely cover us but makes us able. What was in the former nature unable to be just in God's sight is cleansed now by God's grace to merit justification. Nothing about a pile of dung just covered with snow (Luther). Not a hint of Sola Fide.
