What did Santideva, the authour of the Bodhicaryāvatāra, say about the realisabaility of Buddhahood?

In general, the following positions are accepted by Mahayana tenets:

1. We can achieve Buddhahood because we can separate mind from afflictive obscurations and knowledge obscurations since neither is in the nature of the mind.
2. The way to abandon the two is ultimate bodhicitta: the wisdom directly realizing emptiness in the continuum of a bodhisattva. Since we can (a) cultivate bodhicitta and (b) see things the way they are (empty of inherent existence) then we can abandon the two obscurations and actualize the four kayas. There is the following analogy: wisdom is like an ax while bodhicitta is like a strong arm. Having developed both the ax and the strong arm – wisdom and method – one can cut off the tree, the two obscurations.
3. The mind is empty of inherent existence, therefore it is not inherently one with afflictions (from which it can be separated).

With regard to this, the fourth chapter of the Bodhisattvacharyāvatāra, verses 3 – 4 says:

But how can I ever withdraw From what has been examined by the great wisdom Of the buddhas and their children, And has been examined and examined by myself?

If, having made such a promise, I do not accomplish it through action, Then by deceiving all those sentient beings What kind of rebirth will I take?

Gyaltsab Je says that Shantideva speaks here of "The reasons why it is inappropriate to discard bodhicitta". In addition, Geshe Jamphel Gyaltsen explains that:

We increase the capacity of the mind by investigating and understanding. [The mind] has the capacity to develop into anything possible. The capacity of the mind is limitless like the mind of a buddha.

Apart from these verses, Shantideva does not establish that Buddhahood is feasible. He does not expand on any of the previous (three) arguments. Taking them as a given, he shows how to achieve Buddhahood. He does so in a didactic manner. This might be because he addresses people that are already convinced (i.e. in the Mahayana lineage). The Bodhisattvacharyāvatāra belongs to the genre of Mind Training (Lo Jong). Lojong texts include instructions aimed at cultivating bodhicitta, the cause of perfect enlightenment. This is because according to Mahayana, (1) bodhicitta is the entry door to the Mahayana path, (2) bodhicitta is the main cause of Buddhahood, and (3) Buddhahood is the ultimate result of the Mahayana path. As Geshe Jamphel Gyaltsen said:

The Mahayana lineage refers to great compassion, the substantial cause of bodhicitta. [And] The ultimate benefit [of bodhicitta] is the state of complete enlightenment or Buddhahood.

These notions and these arguments are accepted by most Mahayana texts. Therefore, they do not bother establishing that Buddhahood is feasible. However, if you attend teachings on the Bodhisattvacharyāvatāra, or on a Lam Rim text, or on Asanga’s Ornament of Clear Realization, you will probably hear about these arguments (that the two obscurations are not in the nature of the mind, they can be abandoned, that bodhicitta is the cause of Buddhahood, that you need the accumulations of method and wisdom, etc.)

If you want to know more about systematic arguments on the feasibility of Buddhahood, I suggest you investigate other texts. For instance, you could study Tathāgatagarbha (Buddha Nature) by reading:

• When the Clouds Part: The Uttaratantra and Its Meditative Tradition as a Bridge between Sutra and Tantra. Tsadra Collection.
• Great Vehicle Treatise on the Sublime Continuum differentiating the Lineage of the Three Jewels (mahayanottaratantra-ratnagotravibhanga) by Maitreya
• Gyaltsap Darma Rinchen's commentary to Maitreya.

As my teacher says “The ninth chapter is on wisdom and it all leads up to the result we search, Buddhahood.” In chapter 9 of the Bodhisattvacharyāvatāra, you find systematic refutations of non-Prasangika systems (some Mahayana, some Hinayana, some non-Buddhist). Shantideva explains that one could not achieve even arhatship without realizing emptiness of both persons and phenomena. He dismisses so-called Hinayana tenets such as Vaïbashikas and Sautrantikas. It is as close as you will get to finding explanation of the feasibility of Buddhahood in Shantideva’s Bodhisattvacharyāvatāra, but it does not really answer your question.

Concerning “how to achieve Buddhahood”, Gyaltsab Je’s commentary explains that “Buddhahood will not be accomplished by practicing only method or wisdom." The meaning is indeed: one achieves Buddhahood by way of method (bodhicitta and the six perfections) and wisdom (directly realizing emptiness of all phenomena). That is what Shantideva explains.

In terms of view, it is the wisdom realizing the two emptinesses (as presented by Mahayana tenets, namely Prasangika Madhyamika). In terms of method, it is generating the mind of enlightenment as to enter the Mahayana path, cultivate the six perfections, etc. In terms of views, he explains one can not attain liberation is not possible without realizing emptiness of both person and phenomena. Thus, he says that achieving even individual liberation without having directly realized emptiness of phenomena is not feasible. In terms of method, he explains that a Mahayana path is a “virtuous mind conjoined with renunciation and bodhicitta” and that is the cause of buddhahood.

All in all, positing “Buddhahood is not feasible” would imply that we are unable to generate bodhicitta or that bodhicitta is not the cause of Buddhahood (neither is accepted by Shantideva). Or it would imply that we can not accumulate the two collections or that they do not cause Buddhahood (and neither is accepted by Shantideva). Or it would imply that our minds do not have the capacity to abandon all obscurations (that is not accepted by Shantideva). Or it would imply that the mind exists inherently and therefore it is inherently one with the afflicted side (it is not accepted either).