QUESTION: Was Prthu born as a Vishnu-Avatar, or did he become one gradually? (I ask this because the Bhagavatam said he attained his “perfected form” after doing yoga at the end of his life).
A small side point: Pṛthu wasn’t “born.” He was created by sages, from genetic material taken from Vena’s arm. (see 4.15.1) He seems to have been created as an adult, and was immediately coronated King.
As soon as they created him and his female counterpart (Arci), the sages were aware that he was Viṣṇu and she was Lakṣmī. (see 4.15.2-3). Brahma came with all the gods, and confirmed the sages’ opinion via palmistry. (see 4.15.9-10).
So, to answer the first part of your question is that he was a Viṣṇu-Avatāra from the very beginning of his life.
QUESTIONER: Then what does it mean that he “achieved his perfect form” at the end.
Book 4, Chapter 23 describes the end of Pṛthu’s life. I didn’t find the phrase “achieved his perfect form” in that chapter. The general idea of the chapter, however, is that he performed yoga and attained perfection. So, you may be wondering “how did he ‘attain perfection’ if he was already an avatāra? Wasn’t he already “perfect”?
The answer is that perfection can manifest in two senses: “natural” (naisārgika), and “situational” (naimittika). An avatāra, by nature, is mostly situational – it is mostly a response to a particular situation. For example, Pṛthu is a response to the situation caused by the world not having a ruler. So, as Pṛthu Viṣṇu manifests only situationally relevant perfections.
Situational perfection draws upon the natural/inherent perfections, but not necessarily fully. It may ignore some inherent perfections that are not required or not conducive to the situational goal of the avatāra.
So, Chapter 23 is explaining how Pṛthu performed yoga to dissolve his situationally perfect manifestation (as Pṛthu), and attain his intrinsic/natural perfection. By the way, what is really a fascinating avenue for research and contemplation is Pṛthu’s natural perfection is not as “Viṣṇu” (male) but as a Lakṣmi (female). (see 4.20.28)
One other important point…
Avatāras are “situational” manifestations of Viṣṇu, because they all have specific goals to accomplish. One of the goals is almost always showing the right way to live (dharma-samsthāpana). This is why Pṛthu followed a conventional path, retiring to the forest and gradually intensifying a spiritual practice. To show, by his own example, how people should endeavor to attain perfection.