Hello Bsmith, 
I have read through your post and believe I have a good understanding of what you are trying to achieve, also well done for a detailed first post, one might think you have been here before hehe 
First up is the Mac monitor, from my research these displays have only a single video input, which in this case is mini-DP, please correct me if I am wrong in this instance.
A single input on this monitor is of cause a bit limiting, however I see 3 options here that you could choose.
A) You could as you say swap this monitor out for another with additional connectivity, allowing for the pc and mac to be connected at the same time, with a monitor input switch over.
B) You could dedicate the Mac monitor to the PC only, allowing for the HP EliteDisplay to act as your main Mac monitor with its DP input + 3rd PC input via VGA or DVI.
C) You could invest in a Displayport switcher which the Mac and PC output 3 would plug into then the switcher into the single Mac monitor input.
Option B) would allow you to have the 3 monitors you desire with the trade off being that the Mac would make its home on the HP EliteDisplay, however this comes at zero cost to you, as would option A) if you feel you wish to swap monitors out with a co-worker.
Next up is the GPU on the PC.
The PC as you point out has a DVI type connector to 2x VGA outputs.
For just running extended desktops, not an eyefinity/nvidia surround/spanned configuration often used for 3D gaming, it is technically possible to just drop in another GPU, any brand or model would work to give you a 3rd output, so you could pick up something quite cheap.
However note that in practice although this will indeed enable a 3rd monitor output from the PC, using 2 different GPU's (sometimes even identical GPU's) users have sometimes reported issues with poor performance on the 2nd GPU, or stuttering when moving content from display 1 and 2 onto 3.
For a smooth experience it would be advisable to get a GPU that sports 3 outputs, again this can be done fairly cheap if you are not looking for raw 3D rendering horsepower.
I have previously built a video wall using a collection of ATI Radeon 5450's (the ATI... now AMD, 5xxx series was the first series from AMD to offer 3 displays from a single card, in case of nvidia it is the GTX6xx series that offer 3 outputs from a single card), there is of cause also the AMD FirePro, Nvidia Quadro and the Matrox display cards that have for a long time supported 3-4 monitors however these are aimed at workstations running CAD and are generally a lot more expensive than the consumer grade cards.
Although the 5450 worked great for our needs, providing 3 displays from a single card, not using the AMD eyefinity technology.
This card lacks in 3d power, but it was cheap and performed smoothly.
You should be able to find some similar cards with a bit of searching.
One thing to bear in mind is the size of the PSU in the work computer, is it powerful enough and does it have spare PCI-E power connectors for a GPU upgrade?
The 5450 example I gave does not require additional power from a PSU's PCI-E connectors as it is able to draw 70W via the PCI-E lanes on the motherboard which is sufficient for the card to work, as stated, you should be able to find similar cards going cheap and may find something more updated than the 5450, unfortunately I am not in a position to go shopping around for you at this time, however I hope the information provided will lead you in the right direction to resolving your issue.
Last but not least, if you do decide to get a AMD GPU with 3 outputs, then bear in mind that the 3rd output, the displayport output will need to be connected directly to a displayport enabled monitor such as the mac monitor or the HP Elite Display, if you intend to convert the 3rd output, the displayport output, to say VGA/HDMI/DVI then you will require an ACTIVE adapter, passive adapters are also available, but passive will not work, so look out for the active adapters if you choose to not to use displayport to displayport as your 3rd monitor connection method.
I hope this makes sense.
Regards
Delphium. 
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