The Scheming Fence has only gains the activated abilities as long as the permanent is on the battlefield, and it does gain any abilities the permanent gains later.
Scheming Fence's third ability says
Scheming Fence has all activated abilities of the chosen permanent except for loyalty abilities. You may spend mana as though it were mana of any color to activate those abilities.
This is a static ability. Rule 604.2 says
Static abilities create continuous effects, some of which are prevention effects or replacement effects. These effects are active as long as the permanent with the ability remains on the battlefield and has the ability, or as long as the object with the ability remains in the appropriate zone, as described in rule 113.6.
Then rule 611.3a says
A continuous effect generated by a static ability isn’t “locked in”; it applies at any given moment to whatever its text indicates.
So, at any moment, when you're trying to determine what abilities Scheming Fence has, you check what abilities the chosen permanent has at that time. If the chosen permanent doesn't exist, Scheming Fence is not granted any additional abilities.
Scheming Fence gains activated abilities that the chosen permanent gains after it is chosen because of a feature of the layer system called "dependencies". Rules 613.8 and 613.8a say
613.8. Within a layer or sublayer, determining which order effects are applied in is sometimes done using a dependency system. If a dependency exists, it will override the timestamp system.
- 613.8a An effect is said to “depend on” another if (a) it’s applied in the same layer (and, if applicable, sublayer) as the other effect; (b) applying the other would change the text or the existence of the first effect, what it applies to, or what it does to any of the things it applies to; and (c) neither effect is from a characteristic-defining ability or both effects are from characteristic-defining abilities. Otherwise, the effect is considered to be independent of the other effect.
Scheming Fence's ability and Paradise Mantle's ability both grant abilities, so they apply in the same layer. Applying Paradise Mantle's ability first changes what abilities Scheming Fence's ability grants. And neither ability is a characteristic-defining ability. This means that Scheming Fence's ability depends on Paradise Mantle's ability, so Paradise Mantle's ability is always applied first and Scheming Fence's ability always sees it when determining what abilities to grant.