The usual solution to such problems is to launch the program you want to have a special network configuration for in a container, and set that up such that its networking does what you want.
For chrome that's simple, Jessie Frazelle is kind enough to maintain a repository of Docker containers. You can just go ahead and
git clone https://github.com/jessfraz/dockerfiles
cd dockerfiles/chromium
docker build -t jess/chromium
Then, you'd set up a docker network containing your tethering phone and start the container, as specified at the beginning of the chromium/Dockerfile used above (but replacing --net host appropriately).
Alternatively, you could run a SOCKS5 proxy tunnel to your phone, using SSH:
Install termux, install pkg install openssh in there, ssh-keygen -A to generate host keys on your phone in termux. Follow Termux' guide to enable public-key-based access (basically, just copy and paste your laptop's public key via echo "ssh-… " > .ssh/authorized_keys). Start sshd.
Connect to your phone's termux SSH server from your laptop using ssh -D 0.0.0.0:1080 your_phones_local_IP_address. Now, configure your chromium to use a SOCKS server, and use 127.0.0.1 (i.e., your laptop itself, which then tunnels through SSH to your phone), port 1080, by starting chromium with the command line:
chromium-browser --proxy-server="socks5://127.0.0.1:1080" \
--host-resolver-rules"'MAP * ~NOTFOUND , EXCLUDE 127.0.0.1"
#^------+-------^
# \------------------ or just "chromium", depending on your linux distro