Replace Relay with NC Solid State Switch
Build your own solid-state relays (SSR) utilizing optocouplers and Triacs. The tallest part of this build would be the heatsink(s), but you could custom this (i.e. longer over taller) to the space you have available to you. Each SSR can switch up to 12A and based on a 5VDC rail opto supply.

The BT138s can get hot, so you MUST use heatsinks. However, that being said, as you are only running 6-70VAC through them @ an average of 14VAC/8W/1.75A, you will not get as much thermal dissipation from them.
Input Voltage: 5V - 15VDC;
Load Voltage: 6V - 230VAC;
Max Load Current: 12A;
Max Load Power: 2.7Kw