First, it may be possible to study impact of these animals on the epiphytes without necessarily using exclusion fences, for example by comparing populations / communities of epiphytes between areas where these mammals occur and areas where they are absent altogether (for example, the animals may be absent from isolated habitat fragments that may be too small to support a population, which because of their large body sizes, in the cases of pigs, monkeys and deer, typically would require a relatively large 'minimum dynamic area' to maintain presence). But if the mammals are present throughout your whole study area, then exclusion fences may indeed be the best option. Electric fences may be an option for pigs. Else you could consider sturdy metallic fences  that would preclude them from digging their way underneath (for example the fence should project outwards at or just below ground level). For monkeys and rats, a workable and affordable solution could be to have small metallic cages built around randomly selected orchids or orchid clumps that you may then compare with controls (non-caged orchids). Hope this helps.

We used electric fencing to exclude large ungulates (Zebra and Rhinoceros among others) from a region of damaged grazing land in order to rehabilitate the area. If Francois's suggestion is not feasible you could use this method to create exclusion plots. 

François's suggestion (exclusion cages) has actually been used extensively in prairie and alpine meadow ecosystems for grazing evaluation by (relatively) large hebivores (e.g. Mangado et al., Pastagens e Forragens 21: 579-583 (2000); Biondini et al., Ecological Applications 8(2): 469-479 (1998); Bedia & Busqué, Grass and Forage Science 68(2): 297–312 (2013). However I understand that your problem is not really that, as you mention only alien (large) species. If properly set, exclosures would, by default, exclude anything above a certain size. But if you want to differentiate natural from alien fauna, unless that difference is also by size it's gonna be difficult. And, of course, you still have the problem of small grazers that get in. In Pyrenean pastures, for example, grazing by grasshoppers has been found to be on a pair with cattle at certain times of the year. Comparison of data from exclosures with large (no cattle but allowing insects) and small ("no nothing" that walks or flies) mesh size can show this, albeit not without trouble (e.g. shadow effects and others).

And, if the vegetation you want to evaluate is larger (e.g. taller trees), exclosures can be a problem to set up and maintain.

I might suggest a different approach: a posteriori selection. You might perhaps select a (large) number of plots, set up full exclosures for reference and control, but also set automatic camera surveillance. In this way you could evaluate the pressure of the species of interest by recording how long they would be in the plots, and later on correlate the measured density of epìphytes to the number/frequency/length of visits by target grazers on each plot.

HTH. Regards