Yes, I say it's perfectly possible. In fact I've done something similar twice in my life.
1 - At 28 I decided to start cycling to work. I rode for leisure occasionally but this was a big change.
The commute was about 14 km each way and generally uphill on the way to work in the morning and downhill on the way home. The difference in altitude between start and finish was about 140 m so the ride had an overall gradient of about 1%. Note that this felt like a 'proper' challenge at the time
and as much as you might think your ride is about a 5% gradient, that distance/gradient is getting on for a category 1/HC climb (the Alpe d'Heuz is 8% over 14 km) so I'd assume your ride is a good bit gentler than that! You can check altitudes at map locations with tools like this to get a more accurate idea. so your climb will be pretty challenging, fitness wise. (updated after your clarification!)
I was not especially fit or unfit at that time but managed this commute without too many problems. The main one was arriving drenched in sweat until my general fitness had improved a bit.
The bike I used was a GT Palomar - not an especially auspicious (or indeed light!) bike. It lasted well though and I probably put about 3,000 km on it commuting in that time.
It's worth pointing out that I was only using the bike in fair conditions, either taking the bus or car when the rain/snow/ice was too bad.
2 - The second occasion was just recently when, at 41, I decided I'd try and cycle every day for a month. Again, apart from a few leisure rides I'd hardly been on my bike for years.
Unfortunately I picked just about the wettest January the UK has ever seen to try this. I was significantly less fit than I was at 28 and probably carrying about an extra 5 or 6 kg too. I rode almost every day in January, averaging about 10-15 km a day for a total of about 400 km and lost most of the extra weight I was carrying. That I live in quite a hilly area helped as the overall gradient for all my January rides was over 1% (i.e. I climbed over 4,000 m in January).
Initially it was easy as I had a lot of motivation, despite the torrid conditions. Within about a week I was already noticing significant improvements in my segment times but then the weather worsened and it was a real struggle just getting out - there were so many flooded roads - but I persevered. Tracking my improvements helped a lot as I was able to see my personal best times tumble, in some cases I'm now doing some climb segments over twice as fast as I was in January and am in the top 20% on Strava - not bad for an old bloke on a XC MTB!
The bike I used was a Specialized Hardrock, a bit lighter than the GT but still nothing special.
I didn't suffer any real physical problems as I was careful to warm up/down and do lots of stretching. Padded shorts helped avoid any saddle sores. My thighs ached constantly for about five weeks, but I could almost watch the muscle being gained so it's no surprise.
I know that this isn't the recomended way to train and it is sub-optimal if pure fitness/training was the aim, but my goal was more of a holistic 'challenge to myself' and as such I knew what I was doing and that I could get 'fitter' by scheduling in rest days etc.
Now I've scaled my riding back a bit, but my month in the saddle taught me a lot about myself.