Two concerns come to my mind in your case. The first is that, as you note, the beta distribution presumes that there is a non-zero probability of observing values between 0 and 0.5. In that way, the model doesn’t align with what you know to be true about your data.
The second is that, for most sets of parameter values (\alpha and \beta), the domain of the beta distribution is (0,1) (i.e. excluding exactly 0 and 1). So if some of your observations are exactly 1, then the model may not fit at all.
The simplest option that addresses both concerns from a technical perspective is to rescale your data such that 0.5 corresponds to 0 (e.g. 2 \times (x - 0.5)), maybe nudging in the end points so that the actual observed range is [0.0001, 0.9999] (2 \times (x - 0.5) \times 0.9998 + 0.0001), or something like that. However, you introduce some arbitrary-ness through the nudging based on the size of the nudge, so you would want to see how sensitive results are to that choice.
If your data are highly skewed such that there are a large number of 1s, then you might be fine fitting the model as-is (that is, the fitted parameter values might be for a form of the distribution where 1 has a valid likelihood). This might be the case if your data are truly beta-distributed but are so skewed that values below 0.5 just aren’t observed in your dataset. I assume that’s not the case here, but it could be a reasonable assumption if the data are highly skewed.