The quick way to get from where you are to where you want to be.
As we know that bionicpup32 works with your computer, lets stick with that for now.
Boot into your bionicpup32. Just above the taskbar there are icons which provide short-cuts to the drives/partitions. One will have an 'x' on its top-right as soon as you've booted. The 'x' indicate that a partition is mounted/opened and Puppy automatically mounts the partition it's on.
Click that Icon to open the partition, Left-Click an empty space and from the pop-up menu select New>Directory and give it a meaningful name. You'll want one to distinguish it from your current bionicpup32. So perhaps bionic2. After you've created it, Left-Click to open it and leave it open.
A frugal Puppy consists of various files and file-systems. For bionicpup32 you need all those which are colored brown plus the one named vmlinuz in bigpup's post, viewtopic.php?p=62702#p62702. To obtain them as separate, manageable files you may have to again download the bionicpup32 ISO. Once you have that, Left-click it to mount it. From the window which opens, Left-Press, HOLD, then drag the required files into the 'bionic2' folder.
[AFAIK, the lxpupbionic32-4.9.163-pae kernel you mentioned is 'stock' for bionicpup32. If not, let us know and we'll explain how to use it in the Puppy you are building if you need clearer instructions than I've provided below with respect to testing dpup-stretch].
You've just created a frugal install. Now we have to get it on your boot-menu. As you have Windows 7 on the computer, I'm going to assume --for safety sake-- that it is a UEFI computer. While its possible to edit your current boot menu to ADD a listing for the 'frugal bionic' and IIRC bionicpup32 has 'frugalpup installer' on Menu>Setup, I think the easiest way to create a boot-loader would be:
download Grub2Config from here, viewtopic.php?t=3360, click the pet to install it; then Menu>Exit>Restart Graphical Server, to have Puppy recognize that it is now part of your system. Then run it: Menu>Setup>Grub2 bootloader config.
It will write a new boot-loader with stanzas giving you the choice to boot into Windows, your new Frugal Puppy and the 'old' Full bionicpup32.
Reboot into your new bionicpup32. Install the rtl8188fu-lxpupbionic32-4.9.163-pae.pet and again Restart-Graphical Server so that you can setup wifi before shutting down and creating a SAVEFOLDER.
After you reboot into your new Frugal bioncipup32, mount the partition (probably sda1) on which you'll find grub.cfg; then open grub.cfg in a text editor. Add or edit the argument in its 'linux' line to read pmedia=ataflash. Reboot and change the Save Sesssion to 0/zero, then execute a Save. See my post here. viewtopic.php?p=62576#p62576.
All your frugal Puppy files will be in the 'bionic2' folder. So you can delete everything on that partition not in that folder and then edit grub.cfg to delete the stanza for booting into the 'Full install'. But before you delete anything, if you have data files you want to save you can:
mount Puppys partition, Left-Click an empty space and create a folder to store stuff in, e.g., named 'my-stuff'. You can then examine folders in your Full install; then left-press, hold, then drag folders and files into 'my-stuff', select Copy. When your 'cleaning up' the partition, don't delete 'my-stuff'. 
Look in the Additional Software Section, viewforum.php?f=7 for applications such as LibreOffice. But if all you need is a word-processor &/or Spreadsheet, perhaps MikeWalsh's portables of FreeOffice (all or parts) might suffice. viewtopic.php?p=39042#p39042. Similarly, there are various Web-browsers in SFS and portable format you'll find there.
You can, of course, still use Quickpet to obtain applications if you prefer. But SFSes and Portables use little to no RAM when not opened.
Later, if you want, you can explore other Puppys by downloading their ISO, copying their files into a folder as spelled out above and re-running grub2config. As you have a 'troublesome' wifi adapter, it's likely these will also need the rtl8188fu-lxpupbionic32-4.9.163-pae driver and firmware. That means you may have to 'swap' in Bionicpup32's 4.9.163-pae Kernel for the one included in a Puppys ISO.
That's a simple process taking less than 5 minutes as you already have the files you'll need. For example, if you wanted to try dpup-stretch, after copying its 'necessary' files into a folder (e.g. dpup), while still running bionicpup create a folder within dpup to store its original system files. Perhaps name it 'old kernel'. Drag the current vmlinuz and zdrv_stretch_7.5.sfs into the 'old kernel' folder and select move. Then drag the vmlinuz and zdrv_upupbb_19.03.sfs into the dpup folder and select copy. Right-Click the zdrv_upupbb_19.03.sfs, select rename, and rename it zdrv_stretch_7.5.sfs. [The initrd of a puppy has instructions as to what files to include on bootup. For dpup-stretch such files have to include 'stretch_7.5' in their names. For bionicpup32, such files have to include 'upupbb_19.03' in their names.]
Run Menu>Setup>Grub2 bootloader config. You can now reboot, booting into dpup-stretch which is ready to accept and use the rtl8188fu-lxpupbionic32-4.9.163-pae pet.
A frugal Puppy only needs its own folder. You can have as many as your partition can accommodate and you can tolerate and keep track of. Many can make use of the same SFSes and portables. With the exception of SaveFile/Folders, their system files are READ-Only, un-corruptable. You can backup a SaveFile/Folder so that if there's a problem you can switch to the backup. But by only Saving after something has been tested and you're satisfied, there's rarely a problem.
to the "kennels".
You'll realize that the next time some newbie finds his/herself in a similar 'pickle' and we try to locate that post so as not to have to re-write it. 
