You should be able to use a systemic (plant-absorbed, root-killing) product on the ivy without harming the dogwood, as long as it's applied according to its label directions. Most systemics are applied to foliage, where the chemical is absorbed and moved into the plant's roots; they are rarely, if ever, absorbed into roots out of the soil. Sometimes they can be absorbed through bark, but in that case the exposure (a direct spray) has to be pretty saturating and not just a few droplets.

The herbicide active ingredient glyphosate is used in a wide array of products and is pretty inactive and tightly-bound to soil once any contacts the ground. Compared to some other active ingredients, like triclopyr which is more water-soluble and likely to leach (move in groundwater), it's one of the less-risky choices around desirable plants as long as it's applied carefully.

Since English Ivy has waxy, somewhat water-repellent leaves, the addition of a surfactant to the herbicide might be useful. These additives help pesticides work by (in this case) increasing the ability of the spray to adhere to the leaves long enough to be absorbed rather than dripping off right away. Surfactants can also be called spreader-stickers and are sold alongside pesticides at garden centers because they are often combined. Check the herbicide product label first, though, to verify a surfactant can be added, since some formulations might already contain one. The surfactant's own label will detail how to use it.

If you prefer to avoid herbicide, your only recourse is to continue removing all ivy foliage/stems as often as they resprout, and ideally as promptly as possible. The longer the regrowth can photosynthesize each time it appears, the more the root "batteries" of stored energy can be recharged, prolonging the battle of the ivy recovering. While blocking the plant's light might work to weaken or kill it, that can take several weeks or longer, and it might be tricky to find a material that will block enough light while still allowing moisture and air to reach the soil below for the sake of the tree's root health. Plus, ivy can creep out from under a cover, so it's not too low-maintenance since you'd need to patrol the edges of the material regularly to catch and clip any running stems.

Miri

Thank you for your response!

I was wondering if Dawn detergent could be added to roundup as a surfactant? I am using a recipe now containing vinegar salt and Dawn on dandelions and it seems to work pretty well for that.

You're welcome.

We do not recommend the use of Dawn dish soap (or any other household soap or detergent) as a surfactant unless the pesticide label specifically mentions using dish soap. (The pesticide label is the law; it must be used as directed.) We also do not recommend any "home remedy" like vinegar, salt, and detergent. Penn State (in the linked page) provides several good reasons as to why home remedies are risky to use (both for the plants and the environment) and not good pesticide alternatives. They can be unreliable, cause plant tissue damage, or just not work well. Salt (too much sodium) is a toxin for any plant not adapted to seaside conditions, so if it builds up in the soil or leaches into the root zone of nearby desirable plants, it may cause root loss and dieback. Household vinegar is not the same strength as horticultural vinegar (which, due to its caustic nature, is more appropriately referred to as acetic acid), so it not intense enough to cause much plant damage. Soap breaks down the natural waxy layer on plant foliage, which acts like both a sunscreen, barrier to evaporation, and protective coating against fungal spore infection. Damaging or removing it (on a desirable plant) can cause significant leaf and stem injury, though new leaves emerging later (if the roots remain undamaged by the runoff) will replace them.

The mix you're using on Dandelions will not kill the root, only burn the foliage, so they will keep resprouting until the foliage loss is repeated enough to exhaust the root stores. (If an application does kill the root, that must be a lot of salt that surrounding plants will not appreciate.)

Miri

Thanks again! I really don't use the vinegar mixture but for a few places and not in my gardens. Good information!