I have only bought into one core set at the moment. I am considering my next purchase, and given supply issues, I am thinking about buying the latest expansion set Dream Eaters. With only one core for a card pool, will this scenario be impossibly hard on standard level?

They will all be pretty hard with only one core. Ive only played half of Dream-Eaters but so far it seems about average difficulty.

Dream eaters seemed relatively easy to me if I’m honest until Dark Side if the Moon.

I think the bigger issue is that the player cards are maybe not the best for players with fewer card options. But that’s just a feeling I have and not based on much else.

I really wouldn't recommend Dream Eaters as your first cycle. The player cards are quite sparse and somewhat tricky to use properly. If you're worried about "supply issues," both Forgotten Age and Circle Undone have all their packs in stock right now and provide a much broader card pool for a beginner.

It's a tricky question. I agree that The Forgotten Age and The Circle Undone have stronger player cards but I also think that (so far, at least) the scenarios are a fair bit harder in those campaigns too. In particular, I've found the "A" campaign of the Dream-Eaters to be relatively easy. However, being only four scenarios long, the Dream-Eaters campaigns have been generous with experience so you may actually run out of upgrades to buy if your card pool is very limited!

Finally, while one of the packs in each of The Dunwich Legacy and The Path to Carcosa are out of stock direct from FFG it does looks like they are available elsewhere online (although this will depend on where you are located). I'd recommend starting with one of these two cycles if you can, but if you have to go for one of the other three I'd probably recommend The Circle Undone.

In think the investigators in the Dream Eaters tend to bank on a larger card pool more than earlier investigators do--perfectly understandable for the development arc of the game. Other than that, I don't see any problems.

Dream Eaters so far has been easier than the last two cycles. I think you are fine just starting with Dream Eaters. A single core set though is usually a rough place to start. I think you might want to go down to easy if things get too harsh for you. How did NotZ go for you with a single core set? If you were happy with the results there then I think you know what to expect with the Dream Eaters campaign. The Gathering is always going to be a cake walk, but Midnight Masks and Devourers are more taxing scenarios.

I'd recommend looking for a second core set on discount. You can usually find it somewhere for $25 or less in the US. If you had a second core set I'd say confidently you could start any campaign you wanted on Standard. There are some powerful core set cards. The number of cards from the core that are still a strong include make it the best value for your money if you are just looking for an infusion of player cards to fill out decks.