FCC verification for PCB business card?
I can understand your frustration, testing is expensive:

Request quotes from a few test facilities as well as timelines for
when they can get your product into their facility for testing. They
will generally handle the filing after you fill out a few forms.
Quotes for an intentional transmitter may contain sections for the
following (rough ballpark numbers. your results may vary):

CFR 47 Part 15 Section 247 – testing for intentional radiated emissions ~$5k
CFR 47 Part 15 Section 109 – testing for unintentional radiated emissions ~$1k
Test report documents ~$1k
FCC Form 731 and some other filing paperwork ~$2k
IC paperwork – $2k


Source: https://hackaday.com/2016/09/19/preparing-your-product-for-the-fcc/

An unintentional radiator (defined in Section 15.3 (z)) is a device
that by design uses digital logic, or electrical signals operating at
radio frequencies for use within the product, or sends radio frequency
signals by conduction to associated equipment via connecting wiring,
but is not intended to emit RF energy wirelessly by radiation or
induction.
Today the majority of electronic-electrical products use digital
logic, operating between 9 kHz to 3000 GHz and are regulated under 47
CFR Part 15 Subpart B.
Examples include: coffee pots, wrist watches, cash registers, personal
computers, printers, telephones, garage door receivers, wireless
temperature probe receiver, RF universal remote control and thousands
of other types of common electronic-electrical equipment that rely on
digital technology.  This also includes many traditional products that
were once classified as incidental radiators – like motors and basic
electrical power tools that now use digital logic.
Products that only contain digital logic may also be specifically
exempted from an equipment authorization under Section 15.103.

Source: https://www.fcc.gov/oet/ea/rfdevice
The only devices that receive exemptions are these:

Except as otherwise exempted in §§15.23, 15.103, and 15.113,
unintentional radiators shall be authorized prior to the initiation of
marketing, pursuant to the procedures for certification or Supplier's
Declaration of Conformity (SDoC) given in subpart J of part 2 of this
chapter...

Source: FCC CFR 15.101
As stated in the question, your device would not fit in the unintentional radiators exemptions category (15.103) your device is definitely not at 15.113 (Power line carrier system)
It could fit into 15.23 and more on that later:

§15.23   Home-built devices.
(a) Equipment authorization is not required for devices that are not
marketed, are not constructed from a kit, and are built in quantities
of five or less for personal use.
(b) It is recognized that the individual builder of home-built
equipment may not possess the means to perform the measurements for
determining compliance with the regulations. In this case, the builder
is expected to employ good engineering practices to meet the specified
technical standards to the greatest extent practicable. The provisions
of §15.5 apply to this equipment.

Source: FCC CFR 15.23
Your device is an unintentional radiator but not one under section 15.102 (CPU boards and power supplies used in personal computers.) Because these are related to PC's and their components, these components usually are designed for operating inside of a metal enclosure, they also have their own standards that need to be tested.
If your device is not exempted, then you'll need in the least verification because you have a CPU in your product (and it above 1.7MHz check the Frequency allocation list\chart). If your device starts messing around with radio device or knocks someones cell out, someone might complain. It's when people complain (if your radiating on their band) that you get into trouble and become the &quot;responsible party&quot; and if your found to be above the limits, then the producer of the product gets a fine. And rightly so, it can take a significant amount of resources to find an offending device.

RESPONSIBLE PARTIES The FCC has very specific rules as to who is
responsible for regulatory compliance regarding various types of RF
devices. Different rules apply for RF devices subject to various types
of authorization. The responsible party is the exclusive party that is
required to ensure that RF devices under its custody comply with FCC
rules. This includes all identical devices marketed after
authorization. Because the responsible party is liable for
noncompliant RF  devices and related matters, that party will be
subject to FCC enforcement actions in the event potential rule
violations are discovered (see 47 C.F.R. § 2.909).

Source Are Your Company’s Consumer Electronics Exempt from FCC Marketing Regulations?
If it's a prototype then you don't have to worry about testing, but you still have to worry if your device is an unintentional radiator

What does this mean for a hobbyist?
Very little, actually, depending
on what you’re doing. The FCC allows a hobbyist to build up to five
devices of a single design for personal use with no testing
whatsoever. If you are contacted by the FCC (or anyone else) about a
matter of spectrum interference, immediately stop using the device,
don't use it again, and you should be okay. Stick to the ISM bands
(13.56MHz, 27.12MHz, 40.68MHz, 915MHz, 2.45GHz, and 5.8GHz, +/- a bit
for each) for added comfort

Source: https://www.sparkfun.com/tutorials/398
So stick in an unlicenced band for extra comfort and design around that.
Edit -Subassembiles

How do some companies get away without certifying their devices?
You may be asking yourself how companies such as Sparkfun, a business
based on selling electronics kits and wireless development kit
companies continue to sell large numbers of non FCC authorized kits,
with seeming impunity. For Sparkfun, the rules that apply in most
cases relate to “subassemblies”. This just means that Sparkfun’s
customers will most likely use the products to build larger products
containing a number of subassemblies. For example, that may include an
Arduino™ processor board along with several sensors or peripherals and
an LCD. The user may even put all of these parts into an enclosure. If
the user sells this product containing multiple subassembly parts in
an enclosure, for all intents and purposes they are now a
“manufacturer” and their equipment is subject to the normal FCC
authorization procedures.
Where is the FCC’s line in the sand for when home built equipment
becomes subject to their rules? That is defined in Part 15.23.
“Equipment authorization is not required for devices that are not
marketed, are not constructed from a kit and are built in quantities
of five or less for personal use.”
That’s probably easier to understand if I say it in the opposite way;
If you are marketing your product (are you putting out ads, or
offering your product for sale?) or it is not intended for personal
use (i.e. it’s for someone else) or if you make more than 5 of them,
then you need to have your device tested according to FCC rules. If
you do any of these things, the FCC views you as a manufacturer.

Source: http://www.emcfastpass.com/fcc-rules-kits-subassemblies/
So if you want to go the sub assemblies route, then sell them as sub assemblies for use in another product, and don't do any marketing.