Moving a plant that size would be costly and not likely to succeed, at least not without requiring several years for the plant to re-establish itself, likely with some leaf loss in the interim. Complicating matters is the proximity of the roots to the foundation, since the heavy machinery often used for moving heavy, large trees/shrubs can't work in those close quarters. It might also complicate hand-digging root balls (whether attempted yourself or hired-out) if utilities run through that area. Tree and shrub roots generally don't go very deep (the vast majority are in the top foot or so of soil) but spread horizontally further than the branch spread ("drip line"). The more roots you can keep intact when transplanting, the less likely transplant shock will be severe or long-lasting.

If you replace the plant instead, this appears to be a 'Nellie Stevens' variety of holly, very commonly grown hybrid and widely-available selection (and also one of the fastest-growing of the hollies) that you could plant elsewhere for the enjoyment of its bird appeal. If you prefer to use native plants, a better choice might be American Holly (Ilex opaca), which can be very long-lived, but be advised in that case that plants mature even larger than 'Nellie Stevens' but, ironically, are much slower-growing and can be harder to find at garden centers because they don't look as nice in containers when container-grown, even if they're otherwise healthy. Another near-native choice is the cultivar 'Foster's' (Ilex x attenuata 'Fosteri'/'Foster's #2', etc.) which has more slender, less-spiny leaves (well, less spiny than American Holly anyway), a medium-paced growth rate, and a more slender growth habit overall while still reaching two or three stories high eventually. Unfortunately, in recent years Foster's Holly has become a bit hard to find at garden centers also.

Hollies produce male and female flowers on separate plants, and the female plants are the ones producing the berries if they are pollinated. 'Nellie Stevens' is a female variety that happens to produce berries without pollination (one reason it became popular aside from its faster growth). For American and Foster hollies, there are several male and female selections of each, though planting a lone female might be all you need if enough wild male American Hollies grow in the general area where the small bees that visit them can move pollen around. Otherwise, in order to get berries, you'd need the appropriate male (American with American or Foster's with Foster's) in the yard for pollination services. Something to consider, if you're wondering what holly to choose.

You could also severely cut-back the holly as-is, since they tolerate this surprisingly well, though it would of course ruin its aesthetics for the short term while it regrew, and it would also eventually regain its old size so it's not the best solution if it's in the way too often. We don't keep track of landscaping businesses or make recommendations of particular establishments, but you could look around (maybe inquire on social media gardening groups, if you use social media) for landscaping companies that specialize in moving more mature plants and see if they can give you an estimate, or at least a judgement on whether moving this specimen is achievable.

Miri