The use of Vice from the phrase vice versa

Can vice be used in a sentence in the following way. "I would recommend using this product, vice, the other one."

No.

There is a preposition vice, more common in the 18th and 19th century. It is similar to the prefix vice represented in titles like vice president and vice principal, and means (Merriam-Webster):

: in the place of / "I will preside, vice the absent chairman"

Here's another example from the Oxford English Dictionary:

1886 C. E. Pascoe London of To-day (ed. 3) xi. 111 It was..soon afterwards reorganized, with Mr. Randegger, vice Mr. Leslie, as conductor.

In both forms, the preposition is preceded by a comma and takes a (usually personal) object.

So you could technically say "I will use this product, vice the other one" to mean you'll use this product instead of the other one. However, the preposition has dropped out of use in the 20th century except in military contexts, so there's a high risk of being misunderstood.

Furthermore, vice does not serve as an abbreviation of the adverb vice versa. In your version of the sentence, vice versa does not make sense as a synonym for "conversely."

'Vice' in this sense is not an English word. 'Vice versa' is a Latin phrase. It is accepted as part of English only as the whole phrase, and only because it is understood as a phrase.

(Vice is of course an English word with other meanings, but that's not relevant here.)

Not exactly what you ask, but close.

vice PREPOSITION /ˈvʌɪsi/

As a substitute for.
‘the letter was drafted by David Hunt, vice Bevin who was ill’

Oxford Dictionaries