In #2, you can use FX forwards to convert your JPY cashflows to USD but it is more common in practice to use a cross-currency swap for this purpose. Indeed, the advantage of the latter is that it allows you to keep the nominal of your synthetic USD bond constant because the final exchange in the swap is done at FX spot (not forward), and the difference is made up with higher or lower coupons. This way you can use simple yield calculation methods on the synthetic USD bond without issues.
However, the real issue with #2 is that the USDJPY FX Forwards you apply to the JPY cashflows should be defaultable in order to fairly price the equivalent USD amount; if the bond defaults, the remaining JPY cashflows will not be paid and therefore you want your USDJPY forwards you entered into on day 1 to be cancelled, otherwise you will be asked to deliver a JPY amount which you are not receiving from the bond anymore.
The pricing impact comes from the FX-credit correlation. If a Japanese corporate defaults, chances are the Japanese economy is in trouble which should devalue JPY vs USD. This correlation is priced in the quanto CDS market; how much cheaper is a credit protection which pays in the local currency of the entity whose default you are protecting against, compared to a protection which pays you in a hard currency like USD?
Using defaultable FX forwards is the only way this method could make sense.
However, let's take a step back and evaluate your main question: as a USD investor with JPY bonds on my broker account, the price of my bond fluctuates everyday and so does the USDJPY FX. My daily P&L in USD would depend on those two elements. This is what you describe in #1 and I don't see why you would need anything else.
If the question is, how do I protect myself against the USDJPY movement, or how do I replicate such a bond, then you can start exploring cross-currency swaps (defaultable for fair pricing) or look at the corporate CDS as a proxy for the bond's Z-spread. On the latter note, as others pointed out, the CDS-bond basis will have an impact.