As far CLB mode getting engaged after pushing it will not.
The CLB mode can be engaged, if the following conditions are all met:
‐ The aircraft has been in flight for more than 5 s
‐ The selected FCU level is above the present aircraft level
‐ The descent, approach, or go-around phase is not active
‐ NAV mode is engaged
‐ Glideslope (G/S) mode is not engaged.