The scattered brown tips might be due to a soil moisture issue or to a minor pest like caterpillars of the native moth Arborvitae Leafminer. (They are named for their arborvitae preferred host plant, but they can also consume Chamaecyparis.) It sounds like  a sufficient level of direct sunlight (more would be ok, though, as they are tolerant of full sun) but the watering pattern may need to change to keep the plant hydrated without over-watering. Feel the soil about six inches deep next to the roots and water well only when it's becoming somewhat dry to the touch at that depth. The duration of water you're giving the plant now sounds too light, where it won't soak in deeply enough to give the roots much water. For a plant this size, you may need to apply a few gallons per watering session, which can be done with a watering can, bucket with holes in its base that can slowly drip water into the soil, or a soaker hose loosely coiled around the drip line of the shrub. (You can of course water by hand with the hose, too, but ideally should use a water breaker nozzle to break the water stream into a gentler "rain" and it may need to be applied for several minutes to get that amount of water into the root zone. One way to do this without having the excess run off the soil surface is to water lightly, move on to other plants, and then come back to the plant to water again, which gives each watering time to soak in.) Our Watering Trees and Shrubs page has more advice on how and when to water.

If Arborvitae Leafminer is present, they are not major pests and the simplest approach to their damage is to just carefully trim off those brown tips. Pesticide use would be a bit overkill but it's also hard to find a low-toxicity option that will impact any insect feeding inside of foliage as these do, where they are shielded from direct contact with a spray. Spinosad, a leaf-absorbed and naturally-derived ingredient, might work if the population were to boom in a future year. (Bt can work well for young caterpillars like bagworms, but in this scenario, since the caterpillars are inside foliage and Bt is not absorbed into the plant, it would be ineffective.) Arborvitae Leafminer has only one generation per year and eggs are laid around midsummer (July-ish) so if needed, an application made around this time of year or in August sometime may be successful in interrupting that generation. The pictured plant does not look severely infested, though, so for now we would not recommend use of an insecticide. Additionally, this kind of feeding damage can look nearly identical to water stress or root dieback injury to the foliage, so close examination would be needed to confirm the cause of browning was insects and not other factors. Leafminer insects emerge from their feeding chambers through a small hole in the foliage, so if none of the brown tips has a tiny hollow somewhere in that region of damage, leafminer would be less likely.

Hinoki Cypress grow slowly enough, and shrubs in general rarely suffer from nutrient deficiencies, that fertilization should not be necessary. Use caution if fertilizing as over-dosing can stress or injure roots, leading to similar-looking damage of yellowing foliage, brown tips, and a reduction in vigor. Discontinue use of the copper fungicide because a fungal infection is not indicated by the plant's symptoms. (Even though root rot organisms are fungus-like, they are not treatable with a copper-based product and not curable once they set in.) Over-applying copper can also damage plants and fungicides as a group sometimes risk harming other organisms, so always try to confirm the presence of a particular disease before using any sort of treatment. Fungicides are preventative measures and cannot cure existing disease nor reverse its damage.

Overall, the plant appears to be in decent shape, and adjusting your watering approach may improve its condition and tolerance for high heat. (Hinoki Cypress prefer the generally-cooler weather of the Pacific northwest -- where most of those in the nursery trade are grown -- and while they handle our heat admirably, it may tax their vigor a bit.)

Miri

Miri - thank you so much for your thoughtful and in-depth response!

You're welcome!

We don't think that sunlight is related to the browning.
 Drought damage can be slow to show and hard to reverse. It is likely that the plant lost some roots earlier in the season and can't support well all of the foliage it currently has.

They still don't look terrible though, so keep an eye on them.
We don't generally recommend the use of water gauges.
It would be better to check manually, using a screw-driver and your finger to feel the soil. After watering, the soil several inches down should feel cool and moist.
If you have not been able to digest the link given earlier, this page will explain the best way to manually make sure the water is getting where you think it is: 
https://extension.umd.edu/resource/watering-trees-and-shrubs


Christine