Apparently the OEM EVSE is very fussy about the power and does the most saftey checks, the cheaper Chinese EVSEs do less checks and are more apt to overlook power variances. As your portable battery pack has a max output of 2.5kW, of course, you'd want an EVSE that outputs a maximum of less than that or an adjustable one that can be dialed down.
I personally have had very good luck with the cheaper Chinese EVSEs but some don't like them as they don't perform all the safety checks a more traditional EVSE does. Amazon sells a lot of different EVSEs as does other online places note that even though many of the cheaper EVSEs are sold as L2(200-240v) most also work just as well on L1(100-120v) just look on the voltage input rating. Note their output current for L1 and L2 are generally the same, that is if the L2 EVSE current rating is 16a it will be 16a on either 100-120v or 200-240v. Difference being on L2 voltage it will be around 3.5kW and on 120v it would be 1.7kW
This one specifically says L1(110v) https://www.amazon.com/Morec-NEMA5-15P-Charging-100V-120V-Compatible/dp/B089GNZ4R5/ref=sr_1_4?keywords=120v+evse&qid=1636898716&qsid=145-5048073-2441250&s=automotive&sr=1-4&sres=B089GNZ4R5%2CB075GJK2S9%2CB07SKN6V5C%2CB077N25YTG%2CB08RY7S8Z2%2CB07H9H7W2N%2CB07Y7HPKB6%2CB07SKN1BZT%2CB095GY4WT5%2CB08HHBDQ88%2CB08LMX4TJR%2CB07WXZDHGV%2CB08TM1C3BV%2CB08RY1GG68%2CB08Q22651D%2CB09BG18LVS%2CB07DWVNLCZ%2CB085VN8XBH%2CB07HH4LJD2%2CB07V7V2QT8
This one looks to be a L1 and adjustable. https://www.amazon.com/Megear-100-120V-Adjustable-Portable-Electric/dp/B08LMX4TJR/ref=sr_1_17?keywords=120v%2Bevse&qid=1636899147&qsid=145-5048073-2441250&s=automotive&sr=1-17&sres=B089GNZ4R5%2CB075GJK2S9%2CB07SKN6V5C%2CB077N25YTG%2CB08RY7S8Z2%2CB07H9H7W2N%2CB07Y7HPKB6%2CB07SKN1BZT%2CB095GY4WT5%2CB08HHBDQ88%2CB08LMX4TJR%2CB07WXZDHGV%2CB08TM1C3BV%2CB08RY1GG68%2CB08Q22651D%2CB09BG18LVS%2CB07DWVNLCZ%2CB085VN8XBH%2CB07HH4LJD2%2CB07V7V2QT8&th=1
This one seems to be the same as above and only costs a bit more but does both L1 and L2. https://www.amazon.com/Megear-100-120V-Adjustable-Portable-Electric/dp/B07W8ZPZ24/ref=sr_1_17?keywords=120v%2Bevse&qid=1636899147&qsid=145-5048073-2441250&s=automotive&sr=1-17&sres=B089GNZ4R5%2CB075GJK2S9%2CB07SKN6V5C%2CB077N25YTG%2CB08RY7S8Z2%2CB07H9H7W2N%2CB07Y7HPKB6%2CB07SKN1BZT%2CB095GY4WT5%2CB08HHBDQ88%2CB08LMX4TJR%2CB07WXZDHGV%2CB08TM1C3BV%2CB08RY1GG68%2CB08Q22651D%2CB09BG18LVS%2CB07DWVNLCZ%2CB085VN8XBH%2CB07HH4LJD2%2CB07V7V2QT8&th=1
Lots of others on Amazon. I personally kinda like the last adjustable one I linked and looks quite similar to one I have but mine goes up to 30a.
Lastly as you said 110v/50hz I assume you are not in N. America so if the plug ends of any of these EVSEs arent your local standard, you(or someone else knowledgable in such things) will need to cut off the supplied end and wire on the correct end for your local plugs.
Lastly I'm not sure how big(capacity) your portable battery pack is but unless it's very large(heavy) I wouldn't expect it to be able to charge your car for a very long time, just saying.