*Disclaimer* I'm a noob to the Meraki platform, just want to see if I could help somehow with the chain of thought.
I am thinking you could use VLSM to split up your network into the parts you need. For example, using network address 192.168.1.0/24, one could divide it into four subnets of 62 hosts /26 mask. Each subnet will represent a vlan. Under Addressing and Vlan you'd enter the subnet, vlan ID, Name, MX IP and group policy if you have one.
Subnet V_ID Name MX IP G_P
192.168.1.0/26 10 Sales 192.168.1.1 None
Then under DHCP, each vlan you created should come up with the option to configure DHCP for it.
VLAN 10 (Sales)192.168.1.0/26
Client addressing: Run a DHCP server
Lease time: 1 Day
DNS nameservers: Enter your own or use google DNS depending on your needs
Add reservations if required.
Next, set one of the MX's port to Trunk which would be connected to the 1st non-Meraki switch and allow the vlans you created.
While on that switch you would trunk the corresponding link back to the MX, create the vlans and also configure a trunk port to the 2nd non-Meraki switch, trunk the corresponding link back to the 1st switch and create the vlans. just to assign the ports to the respective vlans. Also, set your default gateway to the ip of the MX on both switches.
I hope I didn't miss anything important, but if I did hopefully someone can point it out and we both can learn from the experience.
I think all that is left is to for you, however, is to select the most appropriate netmask for your network's needs and you should be good.