For strict Vaishnavism adherents, Balarama is part and parcel of the Dashavataram.
The Gautama Buddha of Buddhism is not an avatara of Lord Vishnu.
As described in previous posts Lord Vishnu came as Buddha avatara to mislead asuras.
There is valid proofs in Vishnu purana and other puranas that Vishnu had taken Buddha avatara
Another version in Puranas is Lord Vishnu took the form of Buddha to mislead Tripurasura and thereby helped Siva in killing the Asura.
This Buddha is no way related to Gautama Buddha.
In Mahabharata and Vishnu Purana,
Lord Narayana plucked out two strands of hairs. One was black (Krishna) and one was white (Balarama).
Also, as per, Vaishnava texts, Lakshmana and Balarama are expansions of Sankarshana form of the Lord.
So, there is absolutely nothing wrong in counting Balarama as part of Dashavatara.
Now the question remains there are many instances that Balarama is considered the avatara of Adisesha, the primordial devotee of Lord Narayana/Vishnu/Krishna.
The Same Adisesha was Lakshmana in Rama avatara..So, still the question remains why "Balarama" is included in Dashavatara.
One the reasons provided is that the kind of service that Adisesha provides to Lord Narayana/Vishnu/Krishna and as Lakshmana to Rama during Rama avatarama is and was unparalleled. Lord made him his elder brother during Krishna avatara, so that he can serve his elder brother. But, Lord being the Lord and Adi sesha being his primordial devotee, Krishna couldn't match the services provided by Adisesha and serve "Balarama" to the extent, "Lakshmana" served his elder brother "Rama". So, to honor the services of Adisesha, "Balarama" is included in Dashavatara.
I found another excellent answers in one of the posting on Srivaishnvism on why Balarama is part of Dasavathara?
Another reason that is given:
Vasudeva’s first wife was ROhiNi. The second wife was Devaki. For the 7th time, Devaki conceived and it was Balarama. But, Lord’s “YOga Maaya” transferred the Lord Balarama from the womb of Devaki to that of ROhiNi. With this background, we have to see the explanation of Andal (one of the Alwars).
In Verse 17 of Tiruppavai “AmparamE…” while referring to Balarama, she says “Sempon kazhalaDi selvaa, Bala DEva”
Commentators explain that this expression heaps praise on the ‘TiruvaDI’ "the feet" of the elder Balarama. What did this feet do?
Because Balarama was born earlier, he had cleansed the womb of Devaki with his feet to make it fit for the birth of Krishna .
But this was not the case with fate of the earlier 6 children of vasudeva and devaki.
Thus, Lord Balarama saved Krishna by cleansing the Womb of Devaki with his feet. Balarama started serving the Lord even before Lord Avatar took place, unlike Lakshmana who started his service much later (Lakshmana was born much after Lord Rama and started serving him after tha) and earned the sobriquet “LakshmaNO Lakshmi sampanna:
Thus, in recognition of this feat and “MahOPakaara’ of saving Lord KrishNa, Balarama is recognized as an Avataara.