Sensitivity analysis on constants: logit transformation for ANOVA

I have categorical looking-time data (looks to visually presented items A, B, C and D over a number of seconds), containing 0s and 1s. I want to compare groups (adults, children; n=~30 for each group) and conditions (type of auditory input). Looking times to A and B are both of theoretical interest, and so I do not have a single 'target'. I am an SPSS user.

I am not at all well-versed in stats but have tried to use information from this site to inform my approach (also some psycholinguistics papers, e.g. Barr, 2008 and Jaeger, 2008 in Journal of Memory and Language). I believe my best option is a logistic regression -- perhaps multinomial, if I can find a good way to fit it in SPSS. My 0s and 1s are likely generated via the same mechanism as the rest of the data, and so I would perhaps start by trying a fractional logistic regression, as advised elsewhere here.

However, for various non-statistical reasons I may have to go the less-favoured ANOVA route with logit-transformed data. That means choosing a value to add to my 0s and subtract from my 1s. I recognize that choosing a minimally disruptive value is crucial, but other than 'symmetry' on a plot I don't know how to pick values to try, conceptualize the different ways in which 'disruption' might show up, or measure the disruption.

1. What criteria should I use to compare the results from trying out different ways of mapping 0s and 1s to a particular value?

2. The first comment on this question suggests 'At a minimum you would need to do some sensitivity analysis to see the effects of the constants you used' - how would I go about sensitivity analysis?

3. Is there anything in the literature that specifically supports this approach: setting epsilon to half of the smallest non-zero value and replacing all 0 values with epsilon and all 1 values with 1-epsilon? Edited to add: And am I correct in assuming that these values should be calculated separately for my two groups, i.e. for children and adults?