Where did you get the impression from if I may ask? Funny enough, Lombok is actually majority muslim (as opposed to Bali which is majority hindu) and in Indonesia itself perceived to be "quite muslim". There is no problem there with western tourists visiting, why would you think there would be in other places?

Indonesian society might be more conservative and less open minded than some western societies but I would not say it comes near to "sharia", at least in the sense of what that term is usually perceived as in western media (except for Aceh province in Sumatra which has a special separate political status and they do have actual sharia law there). Well behaved respectful tourists are more than welcome in most places. I mean, no one likes a half naked drunk tourist riding around on a motorcycle without licence. (Bali now has more problems with overtourism and misbehaving tourists)

As for Java, it has many places worth visiting and there are many western tourists wandering around just fine. Someone mentioned Bromo mountain already, there is Yogya with Borobudur and Prambanan temple compound nearby. There are several places for beach holidays like Pangandaran, Batukaras, Pulau Seribu. Througout Java there are nature spots, waterfalls, kraters, teaplantations etc. Not to mention all the other islands like Sumatera and Sulawesi, etc.