Hi there! I'm a bit of a noob here, so please forgive me for not knowing most things like others may. I have a router/modem device (which I shouldn't tamper too much with seeing as most of the household is connected to it), and then my OpenWrt device with DHCP disabled, if that matters. I'm in Turkey, by the way.
Tried installing Zapret via this package, but it doesn't seem to work for me.
What do you recommend I do?
Likely start with tor browser
https://www.torproject.org/download/
You cam make tor-ified guest-like network, search openwrt wiki.
It is slow, probably something like xray or passwall has tailored settings for Turkey, unlikely profiles for other places will gain anything.
DPI is by definition a moving target, with both parties constantly tuning the system to fool the other. Therefore a one-fits-all turn-key solution isn't very likely to be possible. at least not unless one of the parties has lost their grip.
What that means for you, is that there is no easy or simple solution that you could forget about after installing it. Nor really that much of advice we could provide, you need to find more local support, who know the situation on the ground and can help you when the knobs are turning (again and again).
Foreign SIM card and a 4G/5G modem. This way, your traffic does not even pass through the DPI, and travels in an encrypted form via a completely separate network into a foreign country.
Yes, very expensive (due to roaming costs), but bullet-proof: works even in China.
Simply use a VPN and configure split gateway:
I spent less than $2 a month for a WireGuard tunnel. My provider provides ~30 servers, both domestic and international. It works great for me with this approach. Won't work in China because they use DPI and recognize WireGuard signature, but you can always try other lesser known types of VPN.
Your edit mentions Zapret, which is a system specifically geared towards bypassing Russian DPI. To be effective, it needs to be tuned according to the weaknesses of a specific DPI implementation (there are multiple).
It is not supposed to be useful against DPI types found elsewhere.
Now that you mention Turkey in yet another edit, one possible option would be a VPN that masquerades as an HTTPS connection, thus thwarting the DPI. That is, an SSTP VPN. There are multiple providers out there, and the most popular ones are likely blocked by IP, so I am not going to recommend a specific one.
OpenWrt support is provided through the luci-proto-sstp package, but beware of a security issue (OpenWrt does not validate the server certificate).
Zapret has a blockcheck.sh file which can be used to tune the settings so that you can successfully circumvent DPI of your ISP. It checks various configurations to check what gets the job done for the blocked domain you typed in, for http, https and tls 1.3/quic. Once its done, you should be able to tailor your installation of zapret using that configuration.