Expensive business model red flags here, main two are your making technical assumptions and burying the lead by glossing over technical requirements.
Normally that's an okay common issue to ignore when someone new is asking questions about software-development on a platform but if your going to be mucking about with regulatory data like a businesses tax info then risk is very different.
I seriously suggest you rethink how your approaching platform integrations to avoid serious issues during either development or in the future.
I was thinking the reason for this could be that Shopify Plus may be required to install private apps
No not how that works , custom apps(private apps are now depreciated) are available on all plans, for Plus plans the only difference is custom apps can be installed across multiple stores owned by the same client(see app types). Which generally only matters if your app model isn't a public app.
As the project lead you should not be making any technical assumption like the above that's an expensive risk.
Your developer(?shopify developer?) should already have discussed this with you in some fashion if you did properly discovery , or be able to do so in discovery.
Making an assumption like that STRONGLY indicates you need to pump the brakes and do discovery with a developer so these types of uncertainties are removed.
And/Or double check that developers experience and understanding of the shopify platform. If the current dev can make the app but not explain shopify very well then go on the experts marketplace and find a developer to do discovery with or at least get a consult to clarify requirements. https://experts.shopify.com/
"Is there a way to bypass the need for Shopify Plus so customers (shopify users) can use my app / software with a lower cost? "
WHAT are you bypassing? But WHY is shopify plus needed? what is the EXACT technical|business distinction that makes "competitors" require a Plus plan.
Just treating something as a barrier to be jumped over leads to glossing over critical implementation detail and that leads to bad & expensive software development projects.
You need to talk to your developer, or do outreach to developers or your competitors sales department to find exactly why|when some competitors have that requirement, then examine how you frame this type of question when you lack such critical detail.
If you don't define that and speak of that exact issue in concrete terms even when fishing for answers you can easily end up with a major business and software development issue because too many costly assumptions and expensive ambiguities were allowed to creep in from the start.
This a business process problem where your not detailing project requirements in advance but expecting them to be formed as you go.
I strongly encourage you to first address methodology before anything else so that going forward you can avoid big pitfalls when doing an integration for any platform not just shopify.
Talk to your dev if you already have one, and/or do discovery|consult with a shopify expert, then introspect >> then do project.
