Custom loss for low false positive rate (higher precision)
You have two options here that I can see:

1) quick and very very dirty - rebalance your data so that negatives are favored in the training data. Then your model will prefer to predict negative rather than positive, and false positives will be suppressed. To see this, consider what happens if all training data were negative. Tune the data balance until you get the desired result (I feel like a bad person for saying this...)

2) use a custom loss function such as the $F_\beta$ score (https://en.wikipedia.org/wiki/F1_score). However, this may be numerically tricky to implement for training as the gradient of the naive function generally behaves badly.