All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
Firstly, it should be noted that there is a difference between "Tawassul", meaning a plea to Allaah by means of an intermediary, and "Istighaathah", meaning seeking relief or calling on someone for help during distress.
As for the story in relation to the Khuzaa‘ah tribe and how they sought help from the Prophet, sallallaahu ‘alayhi wa sallam, when the Quraysh breached the peace agreement and aided the Banu Bakr against them, this is a matter of seeking help (Istighaathah) from a living person with something within his capabilities, and it does not consist in Tawassul. It is permissible for a Muslim to seek help and relief from a living person in times of difficulties and distress; in fact, this is not condemned in the religion, customs or reason. As highlighted in the story, the Khuzaa‘ah sought help from the Prophet, sallallaahu ‘alayhi wa sallam, when the Banu Bakr attacked them with aid from the Quraysh.
However, the chain of narration of the Hadeeth’s version cited in the question contains a weak reporter. At-Tabaraani reported the Hadeeth on the authority of Maymoonah . In fact, we could not find the Hadeeth reported through any other chain of narration in any other collection. It has been reported exclusively by At-Tabaraani on the authority of Maymoonah through that specific chain of narration, and its chain of narration includes Yahya ibn Sulaymaan ibn Nadhlah. At-Tabaraani commented on the Hadeeth saying: “This Hadeeth was reported on the authority of Ja‘far only through (the chain of narration including) Muhammad ibn Nadhlah. It has not been reported on the authority of Maymoonah except with that chain of narration.”
Yahya ibn Sulaymaan ibn Nadhlah Al-Madani has been named a weak reporter by Hadeeth scholars. Al-Albaani said about him after citing the opinions of early Hadeeth scholars of him: “Reports from people like him cannot be cited as evidence if not corroborated, as what Al-Tabaraani said after it provides: "Yahya ibn Sulaymaan related it alone."” Hence, this chain of narration of the Hadeeth cited by At-Tabaraani is weak.
Moreover, when ‘Amr ibn Saalim Al-Khuzaa‘i went to the Prophet, sallallaahu ‘alayhi wa sallam, seeking help, it was reported that he composed a poem in which he called upon Allaah in the opening line; it reads (what means), “O Lord! I am appealing to Muhammad, by the time-honored pact of both our fathers.” When he met the Prophet, sallallaahu ‘alayhi wa sallam, in person, he recited the poem in which he was calling upon the Prophet, sallallaahu ‘alayhi wa sallam, for help. He composed the poem first and then recited it when he met the Prophet, sallallaahu ‘alayhi wa sallam, to seek his help and aid; this incident took place during the lifetime of the Prophet, sallallaahu ‘alayhi wa sallam, and not after his death, and this Companion went to meet the Prophet, sallallaahu ‘alayhi wa sallam, in Madeenah in person and did not call upon him while in Makkah. He mounted his riding beast and traveled all the way to Madeenah to seek his help and relief. If seeking a person's aid in absence were permissible, he wouldn't have made that burdensome journey.
To cite this incident as evidence to support the permissibility of making Istighaathah through the Prophet, sallallaahu ‘alayhi wa sallam, after his death goes beyond the norm, and the sound mind rejects it. If it were permissible for Muslims to call on the Prophet for help, and call upon him after his death, the Companions would have done so at times of difficulty and calamities (wars, famine, drought, etc). It is well-known that the Companions did not do so. Rather, they adhered to the Sharee'ah; whenever a calamity or hardship befell them, they supplicated Allaah, offered the prescribed prayer, like Istisqaa’ (rain-seeking prayer), and did similar prescribed acts at times of hardship and calamities.
Allaah Knows best.