I've gone ahead and switched to an EOIP tunnel and placed the DHCP server directly on the interface and it's working now. Hopefully this won't impact the speed of the link significantly.
It is not clear from your original post whether you use the tunnel also for other than DHCP relay traffic. I suppose you do, so of course there is a difference, the EoIP header has 42 bytes while the IPIP header has at least 20, which, as a minimum, means smaller usable MTU on the complete path when you use EoIP. So the question is what means "significantly".
If the performance impact is unacceptable, you can configure both tunnels in parallel and choose the IP addresses at the ends of the EoIP tunnel in such a way that only DHCP relay requests and DHCP server responses would use the EoIP tunnel, so you wouldn't even need policy routing to separate the traffic (i.e. you dedicate a small private subnet to the DHCP relay - DHCP server communication).
For the curious ones, the reason why I could not do the same in the scenario I've mentioned earlier is that unlike IPIP tunnels, EoIP tunnels use a tunnel identifier which must be unique per device and is common for both directions, so a local EoIP tunnel within a single device cannot be configured.