Skip (most of) it
Whenever I need to have my PCs travel, I simply skip the travel time and move directly to the point. In the past, I've tried to play through travel time, adding encounters and resource counting, but it's always felt kind of pointless, because the challenges the PCs are facing aren't directly related to the goal. The longer I kept it up, the more impatient my players got for "real plot".
In this case, it looks like you're just trying to fill up time. You've said they have enough rations to make the trip, and you don't have a clear goal for the travel itself, beyond going to the location of interest. What's more, you want the PCs to have zero progress during the travel, since you want the PCs to fight the boss with minimal resources. This means that if you play out random encounters, they will either gain levels (leveling up at level 1 is very easy) and resources, which defeats your intent, or they will gain nothing, and the challenges will feel pointless.
The key idea is that table time shouldn't be pointless. You've stated that you want your players to experience combat while traveling, and that the time period is important. None of that requires that you play out the entire trip, though. Instead, only play out the parts that matter. For example, if they face bandits during their trip, you can just play that part, and then gloss over the uneventful days of walking. There's no requirement that table time has to be proportional to in-game time: if something's not interesting, you can simply skip over it.
Now, I'm not arguing that you should skip everything except major plot points. I'm only saying that you shouldn't use filler to pad out in-game time, and that table time should be used intentionally, whether it's for character interaction, exposition, or even a gameplay tutorial. However, if you're looking only for something to fill up time, that's a good indication that you should just skip over that time period.
What is it you want to achieve during your downtime and do your fellow players all feel the same? Much like the famous “shopping episode” I find players can have mixed feelings. Some of my players love to know there will be a session of downtime, others shrug their shoulders seeing it as a chance to relax for a session and have a break from intense playing maybe just socialise and not really play, or even be a session they can ask to skip entirely getting nothing from it.
As a rule of thumb when I run them I reduce all timescales for downtime activities to anywhere from 1/3 to 1/8th the time it should take, this allows those players who want to achieve something to do so without other players feeling they then have to fill weeks or months of game time up with exposition. It also allows my crafting players to make that cool suit of enchanted magical dragon armour in a matter of weeks.
I have also run downtime purely remotely between sessions getting players to email me what they wish to do and what they roll so it doesn’t impact on the table.
I would say talk to your DM and fellow players, and consider offering some of the above alternatives as options to playing your downtime. If the party know you planning on making items for them they may also push for that downtime as well.
Best Answer
Item Bonus to Recall Knowledge
Having a well-rounded level 2 library to look through for reference grants a +1 item bonus to Recall Knowledge according to the Kingdom Building rules.
But it makes sense that libraries could be much bigger than the most basic level 2 option, granting higher bonuses along similar lines to magic items that grant those bonuses (maybe +2 at 7th, +3 at 15th level libraries).
It also makes sense that some libraries aren't well-rounded, but instead narrowly focused or particularly focused on the subject of investigation. In those cases, granting an additional circumstance bonus in the standard range of +1-4 seems appropriate.
One simple way to make downtime and access to libraries matter would be to allow the character to spend 1 day of downtime to learn enough new information to try Recalling Knowledge again about the subject, back at the original DC.
Research Subsystem
There's a substantial set of rules for supporting more involved research if that's your jam, though using it for a one-off research topic like this with only one player interested might bog things down.