Ramakrishna Paramahamsa known as a very well Shakta mystic and yogi. Most of his followers consider him as an incarnation of Lord Vishnu. However, Lord Vishnu is known to be a vegetarian. Ramakrishna Paramahamsa is said to have taken fish. So why would he take fish if he was Vishnu avatara?
Avatarahood has nothing to with eating habits or vegetarianism.
There are various rituals which involve partaking small quantities of meat. We can find description of these in rituals related to yajnas as well as in the grhya and shrauta sutras. So meat eating is not something alien to hinduisn.
One of the famous avatar was Sri Rama. There are plenty of references in the Ramayana about his meat consumption.
suraaghaTasahasreNa maamsabhuutodanena cha | yakshye tvaam prayataa devi puriim punarupaagataa || 2-52-89
devii= “Oh, goddess! Upaagata= After reaching; puriim= the city (Ayodhya); punaH= again; yakshhye= I shall worship (you); suraaghata sahasreNa= with thousand pots of spirituous liquor; maamsa bhuutodanena cha = and jellied meat with cooked rice; prayataa= well-prepared for the solemn rite.”
“Oh, goddess! After reaching back the city of Ayodhya, I shall worship you with thousand pots of spirituous liquor and jellied meat with cooked rice well prepared for the solemn rite.”
samaashvasa muhuurtam tu shakyam vastum iha tvayaa || 3-47-22 aagamiSyati me bhartaa vanyam aadaaya puSkalam | ruruun godhaan varaahaan ca hatvaa aadaaya amiSaan bahu || 3-47-23
22b, 23. muhuurtam samaashvasa= for a moment, be comfortable; tvayaa iha vastum shakyam= by you, here, to take rest, possible; me bhartaa= my, husband; ruruun= stag with black stripes; godhaan= mongooses like [civet-like mammals of the family Viverridae, esp. of the genus Herpestes, Marathi manguus]; varaahaan ca= wild-boars, also; hatvaa= on killing; bahu amiSaan aadaaya= aplenty, meat, on taking; puSkalam vanyam aadaaya= plentiful, forest produce, on taking; aagamiSyati= will be coming [soon.]
“Be comfortable for a moment, here it is possible for you to make a sojourn, and soon my husband will be coming on taking plentiful forest produce, and on killing stags, mongooses, wild boars he fetches meat, aplenty. [3-47-22b, 23]
तां तदा दर्शयित्वा तु मैथिली गिरिनिम्नगाम् ।निषसाद गिरिप्रस्थे सीतां मांसेन छन्दयन् ।। इदं मध्यमिदं स्वादु निष्टप्तमिद मग्निना ।एवमास्ते स धर्मात्मा सीतया सह राघवः ।। (वाल्मीकि रामायण, अयोध्या काण्ड, 96, 1 व 2)
Having shown to Sita the mountain-river Mandakini and gratifying her with meat, Rama sat on the mountain slope. Righteous Rama was seated in Sita's company and remarked saying "This meat is sacred. This is savoury roasted in fire". (Valmiki Ramayan, Ayodha Kand 96:1-2)
Hence, there are no rules that an avatar should not eat meat or should have a particular diet. Avatar is beyond all these aspects.