You are using button cards inside the button card and from my experience templating doesn’t work well on the stacked button (custom fields) The good news is the stacked card doesn’t necessarily need to be a button card. I use a Mushroom Chip card when in this scenario. For yours I used a Mushroom template card.
I tested it and it worked well

type: custom:button-card
variables:
my-timer: timer.maddie_quiettime
styles:
grid:
- grid-template-areas: '"time plus" "time minus"'
- grid-template-columns: 3fr 1fr
- grid-template-rows: 1fr 1fr
custom_fields:
time:
card:
type: custom:button-card
entity: timer.maddie_quiettime
layout: icon_name_state2nd
icon: mdi:bed-clock
show_name: true
show_icon: true
show_state: true
show_label: true
name: Maddie Quiet Time
tap_action:
action: call-service
service: timer.start
service_data:
entity_id: timer.maddie_quiettime
duration: '00:10:00'
hold_action:
action: more-info
card_mod:
style: |
ha-card {
border: none;
margin-top: 0px;
margin-left: 0px;
}
plus:
card:
type: custom:mushroom-template-card
entity: timer.maddie_quiettime
icon: mdi:plus
primary: 2 Min
tap_action:
action: call-service
service: script.add_time_to_timer
service_data:
timer_entity: timer.maddie_quiettime
minutes: 2
icon_color: red
card_mod:
style: |
ha-card {
border: none;
}
minus:
card:
type: custom:mushroom-template-card
entity: light.bed_lights
icon: mdi:minus
primary: 2 Min
tap_action:
action: call-service
service: timer.change
target:
entity_id: timer.maddie_quiettime
service_data:
duration: '-00:2:00'
icon_color: blue
card_mod:
style: |
ha-card {
border: none;
}