This is going to be tough.
To emphasize what's already mentioned upthread, I'd suggest anything near a NJ Transit train station. Turn on google maps transit layer and search hotels and find options within budget.
It will not have a NYC "urban vibe" or nightlife around but it's likely to be clean and convenient to get to all of your destinations in 60-90mins by rail. Ideally you can reserve a refundable rate, to change if something better appears last minute.

I haven't stayed here but this BW / Westfield inn is the type of place I'd be considering as it meets all my criteria above, and the rate is $330 refundable right now for your dates.
https://www.bestwestern.com/en_US/bo...oms.31007.html