Did he ever felt sorry for killing Ravana?
Yes, this is described in PADMA PURANA: PATALAKHANDA: Chapter 8. Agastya Advices Rama to Perform Horse Sacrifice.
When Agastya narrated Ravana's story, Rama felt very much sorry about killing Ravana born in a brahmana-family. Agastya tried to convince Rama that Ravana was a servant in Vaikuntha and he obtained demonhood due to the curse of sages. By killing him Rama have favored that brahmana (viz. Ravana) and said Rama to not again grieve over (the death of Ravana).
Sri Rama said :
24-26. Sin is said to be of two kinds, viz. deliberate and nondeliberate.
Deliberate is that which is done with an intention.
The non-deliberate is that committed unintentionally. The deliberate
sin is exhausted only after experiencing (the fruit of that
sinful) deed. The other one would perish through remorse. This
is declared by the scriptural injunctions. The words of the good
cannot be properly regarded as removing the grief of me who
have intentionally committed the extremely censurable killing of
27-28. Tell me a vow, a gift, a sacrifice, a (visit to a) holy
place, or some great worship like that, which would burn my sin,
and by which my spotless fame will purify the people who are
defiled by sinful conduct and whose lustre is taken away due to
the killing of a brahmana.
Then, Agastya spoke these words to Rama
29-36. "O Rama, O great hero, O you who does favour to the
worlds, listen to words which I (shall now) speak for removing
(the sin due to) killing a brahmana. He who would perform a
horse sacrifice, overcomes all that sin. Therefore, O soul of the
universe, you perform that auspicious horse-sacrifice.....
Agastya also told that how others like Indra overcome from Brahmhatya by performing horse-sacrifice.
And then, Lord Rama performed an Asvamedha yajna which is described in subsequent chapters of Padma Purana.
As per Valmiki Ramayana also, Laxamana gave a strong reason (that Indra purified himself from the sin of Brahmhatya by Asvamedha) to perform Asvamedha which indirectly might mean to perform Asvamedha to remove the sin for killing Ravana. This is discussed in Why did Lord Rama perform an Ashwamedha Yajna? post.