TunnelBear vs Private Relay

VPN and Private Relay have different objectives.

A full VPN, without split-tunnel, will pass all your network traffic via the VPN encrypted tunnel as far as the VPN Gateway - where your traffic is delivered to the internet and routed to its intended endpoint.

Private Relay is intended to hide/obscure your IP Address. This support page provides an outline as to how this technology works - and precisely what it is designed to protect:

About iCloud Private Relay - Apple Support

A VPN connection carries a protocol and processing overhead - that will reduce effective throughput and performance, although for a “fast” internet connection, you are perhaps unlikely to notice any practical performance impact. On low speed connections, the performance deficit is more likely to be noticeable.

A VPN connection will effectively replicate any benefits provided by Apple’s Private Relay; as such, attempting to use both in tandem is perhaps unlikely to provide significant technical benefit.

Perhaps consider that for most Commercial VPN Clients, the VPN is essentially intended to maintain privacy over the least-trust portion of your network path - this being the WiFi and associated network with which you make your connection to the internet.

Only some of your network traffic is fully encrypted; some protocols, such as DNS, are natively unencrypted and can be intercepted, monitored and maliciously manipulated. Both VPN and Private Relay will protect this element of your network traffic, however, a properly configured VPN will protect all of your tunnelled traffic.