A biopoem is a style of poetry not found in Shakespeare's writings, so I think instead of "find" a biopoem, you will need to write a biopoem. I'll give an overview of what a biopoem is and of the little--the very little-- we know about Snout in A Midsummer Night's Dream and then you can put the two together and write a biopoem for Snout. A biopoem is a biographical poem that follows a specific form. ReadWriteThink.org presents the biopoem form as a 10 line free-form poem beginning with the subject's or character's first name and ending with their last name. In between is biographical information.
Line two holds adjectives that describe the subject or character. Line three states important relationships, which may include those of sister/brother or son/daughter etc. Line four presents the things loved by the subject or character, including ideas or things. Line five describes the subject's feelings. Line six describes his/er fears.
Line seven lists their accomplishments in a "who wrote ... who said ... etc" format. Line eight lists what the subject or character hoped would happen or hoped they would experience. Line nine tells where they lived, their residence. That brings you back to line ten and the subject's or character's last name.
The biopoem form is somewhat variable. Some forms make it eleven to thirteen lines instead of ten. These forms trade what the subject or character accomplishes for what they give and add what they need, share and are. Applying this to Snout will be fairly simple but also fairly difficult. It will be simple because there isn't that much information to juggle through. It will be difficult because ... there isn't that much information.
Snout's first name is Tom, last name, Snout. These are biopoem lines one and ten. He is unobtrusive, quiet, prone to asking questions, thoughtful to a fault. He likes acting, Peter Quince and Bottom. He feels concern for not frightening the ladies in the audience. He fears giving cause for the ladies to have fears of lions.
Snout has accomplished getting himself into a theater group and being requested to play a noticeable role. He would like to have a wall. He lives in Athens. And again, his last name is Snout. Here follows all nine of Tom Snout's lines:
I,2,320: Here, Peter Quince.
III,1,831: By'r lakin, a parlous fear.
III,1,843: Will not the ladies be afeard of the lion?
III,1,850: Therefore another prologue must tell he is not a lion.
III,1,864: Doth the moon shine that night we play our play?
III,1,877: You can never bring in a wall. What say you, Bottom?
III,1,932: O Bottom, thou art changed! what do I see on thee?
V,1,1998: In this same interlude it doth befall
That I, one Snout by name, present a wall;...
V,1,2049: [as Wall] Thus have I, Wall, my part discharged so;
And, being done, thus Wall away doth go.