Other people have asked you to cover your ground rules already, so we know that whatever you put on can be damaged or removed if the aggressor has access to the components.
I don't know how much ROM data you need. But looking at what can be programmable by you and then stay 'tamper-proof', I'd suggest using a Lattice iCE40 FPGA such as an iCE40LP1K. It contains OTP configuration Flash, so you alone can programme it but no-one can read it or reprogramme it.
As per p6 of Lattice TN1248 'iCE40 Programming And Configuration':
"The NVCM contents are entirely contained within the iCE40 device and are not readable once protected by the one-time programmable Security bits. Furthermore, there is no observable difference between a programmed or un-programmed memory cell using optical or electron microscopy. The NVCM memory has a programming interface similar to a 25-series SPI serial Flash PROM. Consequently, it can be programmed using Diamond Programmer (version 2.2 or later) before or after circuit board assembly or programmed in-system from a microprocessor or other intelligent controller."
You then design an FPGA circuit to implement a ROM of your data and your communications port. That circuit is then loaded into the FPGA configuration from the internal OTP memory.
Your ROM communications port protocol is of your choice. It can be to a standard (e.g. SPI, I2C) or a made-up one that's difficult to understand by an intruder with a 'scope. I appreciate that obfuscation of data will not save you in your application but it will slow those intruders down a bit and its a cheap option.
Which device you need and whether an iCE40 can hold all your data depends how much you've got and I don't know that. Oh, and I think the iCE40 also has an internal oscillator but I'd need to check that.
Putting a conformal coating on to make it very difficult to observe pin activity seems a cheap option afterwards, if its of use. You can either have it done professionally or get medieval with a packet of Araldite.