AVE above 0.5 is very important. Improvement through modification indices is possible only for minute deviation using modification indices that too a certain limit, say one or two in a model. If the value is very less from 0.5 e.g. 0.46 / 0.44 then there is data issue. If by deleting items you can improve the AVE then great (without loosing content validity) otherwise... You need to check the data discrepancy. Respondents may not have filled the responses correctly or check the scale and critically analyze it. Is it really a reflective scale?
You need to look upon these aspects critically. As you said C. R. was OK which should be above 0.7. But AVE is a compulsive parameter, which should be established. Look on these aspects.
For discriminant validity, please go n check your variables (scale's items) in which issue is appearing. There might be a possibility at scale level. If not, then indeed it's a case of data.
I hope it helps.
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