In some tellings of Typhon's revolt, the gods take animal forms and flee to Egypt in panic. The version in the Metamorphoses of Antoninus Liberalis references Nicander of Colophon and explicitly mentions the animals each god transformed into:
[28. Typhon] He felt an urge to usurp the rule of Zeus and not one of the gods could withstand him as he attacked. In panic they fled to Aigyptos (Egypt), all except Athena and Zeus, who alone were left. Typhon hunted after them, on their track. When they fled they had changed themselves in anticipation into animal forms. Apollon became a hawk [i.e. the Egyptian god Horus], Hermes an ibis [the Egyptian god Thoth], Ares became a fish, the lepidotus [Egyptian Lepidotus or Onuris], Artemis a cat [Neith or Bastet], Dionysos took the shape of a goat [Osiris or Arsaphes], Herakles a fawn, Hephaistos (Hephaestus) an ox [Ptah], and Leto a shrew mouse [Wadjet]. The rest of the gods each took on what transformations they could. When Zeus struck Typhon with a thunderbolt, Typhon, aflame hid himself and quenched the blaze in the sea.
Source: ANTONINUS LIBERALIS. Metamorphoses. Translation by Celoria, F. London: Routledge.
This is an etiological myth, the Greeks used this story to explain the origin of the therianthropic deities of Egypt. In the quote, the translator helpfully indicates the Egyptian deity each animal corresponds to. Except for the fawn, the animal Heracles transformed to.
The one Egyptian deity I could find that is associated with deer is Satis. However, a hunting and fertility goddess doesn't seem a likely equivalent of Heracles.
Why does Heracles transform to a fawn?
The original quote is:
[28. Τυφών] οὗτος ἐπεθύμισε τοῦ Διὸς ἔχειν τὴν ἀρχὴν καὶ αὐτὸν ἐπερχόμενον οὐδεὶς ὑπέμενε τῶν θεῶν, ἀλλὰ δείσαντες ἔφυγον πάντες εἰς τὴν Αἴγυπτον, Ἀθηνᾶ δὲ καὶ Ζεὺς ὑπελείφθησαν μόνοι. Τυφὼν δ΄ ἐκ ποδὸς ἐδίωκεν. οἱ δὲ προμηθείᾳ διέφυγον ἀλλάξαντες εἰς ζῷα τὰς ὄψεις· καὶ Ἀπόλλων μὲν ἐγένετο ἱέραξ, Ἑρμῆς δὲ ἶβις, Ἄρης δὲ λεπιδωτὸς ἰχθύς, Ἄρτεμις δὲ αἴλουρος, τράγῳ δὲ εἰκάζεται Διόνυσος, ἕλλῳ δ΄ Ἡρακλῆς, βοῒ δ' Ἥφαιστος, μυγαλῇ δὲ Λητώ, καὶ ὡς ἕκαστος ἔτυχε τῶν ἄλλων θεῶν μετέβαλε τὴν ὄψιν. ἔπει<τα> δὲ Τυφῶνα Ζεὺς βάλλει κεραυνῷ· καιόμενος δὲ ὁ Τυφὼν ἔκρυψεν ἑαυτὸν καὶ ἠφάνισε τὴν φλόγα τῇ θαλάσσῃ.
Source: Μεταμορφώσεων συναγωγή. (2017, Απρίλιος 21). Βικιθήκη. Ανακτήθηκε 07:38, Σεπτέμβριος 15, 2017 από https://el.wikisource.org/w/index.php?title=%CE%9C%CE%B5%CF%84%CE%B1%CE%BC%CE%BF%CF%81%CF%86%CF%8E%CF%83%CE%B5%CF%89%CE%BD_%CF%83%CF%85%CE%BD%CE%B1%CE%B3%CF%89%CE%B3%CE%AE&oldid=94661.